Matter-wave interferometry with composite quantum objects

M. Arndt, N. Dörre, S. Eibenberger, P. Haslinger, and J. Rodewald

University of Vienna, Faculty of Physics,

VCQ, Boltzmanngasse 5, 1090 Vienna, Austria

K. Hornberger and S. Nimmrichter

University of Duisburg-Essen, Faculty of Physics, Lotharstraße 1, 47048, Germany

M. Mayor

University of Basel, Department of Chemistry, St. Johannsring 19,

Basel, Switzerland and Forschungszentrum Karlsruhe,

Institute for Nanotechnology, P.O.Box 3640, 76021 Karlsruhe, Germany

Abstract

We discuss modern developments in quantum optics with organic molecules, clusters and nanopar-ticles – in particular recent realizations of near-field matter-wave interferometry. A unified theoreticaldescription in phase space allows us to describe quantum interferometry in position space and in thetime domain on an equal footing. In order to establish matter-wave interferometers as a universal tool,which can accept and address a variety of nanoparticles, we elaborate on new quantum optical elements,such as diffraction gratings made of matter and light, as well as their absorptive and dispersive interac-tion with complex materials. We present Talbot-Lau interferometry (TLI), the Kapitza-Dirac-Talbot-Lauinterferometer (KDTLI) and interferometry with pulsed optical gratings (OTIMA) as the most advanceddevices to study the quantum wave nature of composite matter. These experiments define the currentmass and complexity record in interferometric explorations of quantum macroscopicity and they opennew avenues to quantum assisted metrology with applications in physical chemistry and biomolecularphysics.

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CONTENTS

I. Introduction and outline 4

II. Concepts and tools of coherent nanoparticle manipulation 7A. Coherence preparation 7B. Far-field diffraction at a nanomechanical grating 8C. Optical gratings 8

1. Measurement induced absorptive gratings 92. Optical phase gratings 9

D. Matter-wave interferometry in the time domain 10E. From far-field to near-field diffraction and near-field interferometry 11

III. A unified phase-space description of three-grating matter wave interferometry 13A. The Wigner function representation 13B. Grating diffraction in phase space 14

1. Thin stationary gratings for fast particles 152. Short ionizing grating pulses 153. Classical pendant of the grating transformation 16

C. The Talbot self-imaging effect 17D. Talbot-Lau interference in phase space 19

1. Coherent description 192. The KDTLI setup 223. The OTIMA setup 23

E. The influence of environmental decoherence 23F. Spontaneous collapse models 24

IV. Talbot-Lau interferometry 26A. Protection from collisional and thermal decoherence 26B. Quantum-assisted deflectometry 28

V. Kapitza-Dirac-Talbot-Lau (KDTL) interferometry 30A. Experimental results: high-mass quantum interference 30B. Experimental results: quantum-interference assisted metrology 34

1. Optical polarizability 342. Static polarizability 353. Vibration induced electric dipole moments 354. Permanent electric dipole moments 37

VI. OTIMA interferometry 39A. Experimental design and conditions 40

1. Requirements on mirror flatness and spectral purity of the grating lasers 402. Vacuum requirements 423. Vibrational isolation 424. Alignment to gravity and the rotation of the Earth 435. Beam divergence 43

B. Experimental results 431. Quantum interference seen as a mass-dependent modulation of cluster

transmission 442. Interference resonance in the time domain 45

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3. Interference pattern in position-space 45

VII. Perspectives for quantum delocalization experiments at high masses 48

Acknowledgments 50

References 50

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I. INTRODUCTION AND OUTLINE

Macromolecule interferometry builds upon the ‘shoulders of giants’, on many ideas frommatter-wave physics with electrons [1], neutrons [2] and atoms [3] which have been developedover the last decades (Fig. 1). The birth of quantum coherence experiments with molecules maybe dated to the early days of Estermann and Stern in 1930 [4], who diffracted H2 moleculesat crystal surfaces. Pioneering experiments with molecular beams based on high-resolutionRamsey spectroscopy of SF6 [5] and I2 [6] which may also be interpreted as de Broglie waveinterferometry [7].

Dedicated molecule diffraction experiments at nanofabricated material gratings started in1994 and enabled proving the existence of the weakly bound He dimer [8]. Shortly later, a three-grating interferometer was applied to measure the collisional properties of Na2 [9]. Quantumcoherence experiments with hot macromolecules were established in 1999 when we studieddiffraction of the fullerenes C60 and C70 [10]. At an internal temperature of 900 K or higher theseobjects share many properties with those of little rocks: With almost 200 internal vibrationaland rotational states hot fullerenes can be seen as carrying their own internal heat bath andwith increasing temperature they show many phenomena known from macroscopic lumps ofcondensed matter: they are capable of emitting thermal electrons, they emit a black-body-likeelectromagnetic spectrum and they even evaporate C2 subunits when heated beyond 1500 K[11]. Yet, on a time scale of several milliseconds, the internal and external degrees of freedomdecouple sufficiently not to perturb the quantum evolution of the motional state. Decoherence-free propagation can thus be established, as long as the internal temperature [12] and the residualpressure [13] are small enough.

The fullerene experiments triggered a journey into the world of quantum optics with complexmolecules, whose rich internal structure is both a blessing and a curse: On the one hand, com-plexity is interesting and it turns out that quantum interferometry provides a well-adapted toolfor studying a large variety of internal molecular properties, with the potential for much betteraccuracy than classical experiments. On the other hand, many developments are still necessaryto prepare ever more complex molecules in pure quantum states of motion. This is particu-larly challenging since only very few techniques from atomic physics can be carried over to thehandling of macromolecules whose internal states often cannot be addressed individually.

Before diving into the details of quantum manipulation techniques for macromolecules, letus discuss what makes nanoparticle interferometry such an interesting and thriving field. Thereis a number of answers to this question with two of them providing the key motivation forour current research. The experiments are driven mainly by the question whether quantumcoherence can be maintained at high mass and in the presence of environmental disturbances[23]. Indeed, one might have doubts whether macromolecules can be prepared and kept in asufficiently coherent state for de Broglie interference. One reason lies in the rich dynamics ofinternally excited macromolecules which may lead to internal state changes on the picosecondtime scale and to increased interactions with the environment over milliseconds to seconds,depending on the particle structure and temperature.

Meanwhile the original concept of decoherence theory [24–27] was confirmed experimentallyin a number of settings [12, 13, 28]. In modern matter-wave experiments decoherence can bekept well under control, and we expect that this should still be the case with particles in themass range of up to 107 − 109 u using present day technology [29–31].

Yet, one may raise the fundamental question whether the linearity of quantum mechanicsholds on all mass scales, or whether it breaks down at some point on the way to the macroscopicworld, a view denoted as macrorealism [32]. One particular example of a macrorealist theoryis the model of continuous spontaneous localization (CSL), which was introduced to solve the

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1923 De Broglie hypothesis

1927 Electron1930 He atoms & H2

1936 Neutron

90‘s I2, He2, Na , 2Alkali atoms

1995BEC1999 FullereneC60

2013m > 10.000 u 810 atoms

Figure 1. Matter-wave interferometry in a historic context. Starting from de Broglie’s hypothesis in1923 [14] the quantum wave nature of matter was demonstrated for electrons [15], atoms and diatomicmolecules [4] and neutrons [16]. Modern breakthroughs in laser physics, nanotechnology, and lasercooling led to the development of atomic matter-wave experiments [7, 17–19], and to studies with quan-tum degenerate coherent ensembles (BEC) [20, 21]. Macromolecule interferometry started with fullerenediffraction in 1999 [10] and led to quantum interference with the currently most complex organic parti-cles [22].

quantum measurement problem and to restore objective reality in our everyday world [33, 34].It is a consistent nonlinear and stochastic modification of the Schrödinger equation compatiblewith all experimental observations to date. It predicts the wave function of any material parti-cle to spontaneously localize with a mass dependent rate to an extension of less than 100 nm.Other suggestions to modify the Schrödinger equation involve gravitational effects, which mightbecome relevant beyond a certain mass scale [35–37].

Such fundamental issues aside, quantum interferometry of complex nanoparticles is develop-ing into a metrological tool. The ultra-short length scale set by de Broglie interference can be usedto measure particle properties by letting them interact with external fields [38–40]. Quantum-enhanced molecule and nanoparticle metrology may well become an important method, in par-ticular with regard to the biomolecular world where increased knowledge of particle propertieshas an immediate relevance.

Molecule interferometry may also lead to future applications not treated here [41]. For in-stance, the molecular interferograms, which are formed during free flight, may be captured ona clean screen, constituting the basis for quantum-assisted molecule lithography. First steps inthis direction have been taken in our labs recently [42].

In all experiments described below, what matters is the dynamics of the center-of-mass mo-tion. It is the total mass of all atoms in the molecule which determines the de Broglie wavelengthand the interference fringe separation. In contrast to that, atomic Bose-Einstein condensates

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(BEC) are described as a coherent atomic ensemble, whose properties are similar to that of alaser. A laser does not change color when the number of photons is increased; similarly, conven-tional BEC interference is described by the de Broglie wavelength of the individual atoms. Onlyrecently, a number of intriguing experiments on BEC squeezing rely on entangled many-particlestates [43, 44].

The structure of the remaining text is as follows: Section II presents an overview of differentdiffraction and interferometer techniques for complex particles. Section III contains a unifiedtheoretical phase-space description of near-field matter wave physics, which serves as a commonframework for all following interferometer implementations. The Talbot-Lau (TL) interferometerwith nano-fabricated gratings is discussed in Section IV. In the following Section V a descriptionof the Kapitza-Dirac-Talbot-Lau (KDTL) interferometer with a central optical phase grating isgiven. An all-optical time-domain matter-wave (OTIMA) interferometer with pulsed ionizationgratings is then discussed in Section VI. We conclude in Section VII with a perspective on futurequantum experiments with high-mass composite materials.

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II. CONCEPTS AND TOOLS OF COHERENT NANOPARTICLE MANIPULATION

This section introduces quantum manipulation techniques for nanoparticles. With the termnanoparticle or molecule we will designate objects in the size range between atoms and 100 nm.We will emphasize the common principles here, leaving the technical details to the following sec-tions. To make matter-wave interference work, we need to establish coherence in the first place.We need efficient beam splitters and interferometer arrangements which fit existing molecularbeam sources, characterized by short de Broglie wavelengths, poor coherence, and low particlefluxes.

A. Coherence preparation

The degree of motional quantum coherence can be defined as the normalized correlationfunction between probability amplitudes at different points in space and time. It describes theability of a particle ensemble to show matter wave interference, i.e. the extent to which onecannot predict the path individual particles will take in an interferometer arrangement.

The molecules are not required to be coherent with respect to their internal states. This isfortunate since it is almost impossible in practice to prepare two molecules in the same internalstate. This even holds for a few-atom molecule where the density of states grows rapidly withthe number of atoms. Quantum interference can always be established as long as the internalstate dynamics remains uncorrelated with the state of motion at all times.

Moreover, also the motional state of the ensemble can be mixed. It is often tolerable tohave a distribution of velocities v and masses m in the ensemble, if the associated de Brogliewavelengths λdB = h/mv agree to within around 10%.

As regards the coherence properties, similar rules hold for matter waves as for light fields:The extension of the beam source, as well as its distance to the diffraction elements, determinesthe transverse spatial coherence. The width of the wavelength distribution on the other hand,governs the longitudinal (or temporal) coherence of the matter waves. Mathematically, these re-lations are engraved in the van Cittert-Zernike theorem and the Wiener-Khinchin theorem [45].The spatial coherence function is determined by the Fourier transform of the intensity trans-mission function of the source aperture, whereas the longitudinal coherence length is inverselyproportional to the wavelength or velocity spread.

Spatial coherence can also be related to the quantum uncertainty principle, providing anintuitive picture. The smaller the source size the larger is the momentum uncertainty of theparticles. As a rule of thumb, we may take the coherence width behind a spatially incoherentbut monochromatic source of aperture diameter D to be determined by the distance between thefirst minima of the diffraction lobe under plane wave illumination. We can therefore estimate thespatial coherence width Wc to grow with the distance L behind the source as: Wc ' LλdB/D.

The longitudinal (temporal) coherence length Lc is a measure for the spectral purity of thebeam, given by Lc ' λ2/∆λ [45]. Different prefactors of the order of 2π can be found in theliterature, depending on the assumption made about the shape of the wave packet, as well asthe definition of its width. Since most macromolecular interference experiments so far were per-formed with thermal beams, the Maxwell-Boltzmann velocity distribution sets a typical limit tothe initial longitudinal coherence of about Lc ' 2λdB . The observation of Nth-order interferencerequires a coherence length of at least N · λ. This can be achieved by using slotted disk [46] orhelical velocity selectors [47], time-of-flight measurements, or by exploiting gravitational free-falland a selection of ballistic parabolas [48, 49].

Since the de Broglie wavelength in nanoparticle interferometry is usually smaller than 10 pm,

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it is desirable to improve the coherence parameters by preparing a very small and very coldbeam source. The loss-less preparation of micron-sized sources can be realized by covering thethermal source with a tiny slit or by evaporating the desired material only locally using a laser-micro-focus evaporator [50]. Motional cooling, however, is still a big challenge. Over the lastdecades, laser manipulation and trapping techniques for atoms have progressed to a level thatit is nowadays possible to routinely generate ultra-cold coherent atom ensembles. In contrastto that, cooling of free nanoparticles to less than the temperature of a cryogenic buffer gasremains a true challenge. First experiments on optical feedback cooling [51] and cavity coolingof nanoparticles [52, 53] have recently emerged for particles in the size range between 70 nm andseveral micrometers. It is however still an open question how to extrapolate these achievementsto objects in the mass range of 103 to 107 u and in an ultra-high vacuum environment. Allmolecule diffraction and interference experiments presented here have been performed withconventional beam sources, based on thermal evaporation or sublimation of the material, insome cases followed by adiabatic expansion in a dense seed gas to favor the formation of clustersand to reduce the longitudinal velocity distribution.

B. Far-field diffraction at a nanomechanical grating

The clearest demonstration of the quantum wave nature of matter is provided by diffractionof particles at a single mechanical grating. We will therefore start by analyzing the requirementsfor grating diffraction of massive matter beams [10, 50]. The angular separation of the diffractionpeaks behind a grating of period d is approximately given by sin θdiff = n · λdB/d, in full analogyto classical wave optics. To resolve the diffraction fringes in the far-field the diffraction anglemust be greater than the collimation of the molecular beam, θdiff ≥ θcoll. Signal constraints,as well as the onset of van der Waals interactions between the molecules and the collimatorslits, suggest that the particle beam collimation should not be reduced below θcoll ' 5µrad. Athermal molecular beam of particles in the mass range of 1000 u requires high temperatures (500-1000 K). It therefore operates typically at a most probable velocity of v =

√2kBT/m ' 200 m/s

with de Broglie wavelengths around λ = h/mv ' 2 − 5 pm. A tight collimation is needed tofulfill the coherence requirements and for a de Broglie wavelength of 2 pm the source widthhas to be as small as 5µm to obtain transverse coherence of 200 nm in 50 cm distance behindthe source. These tight constraints imply already the use of diffraction structures with a periodaround d = 100 nm. Nanofabricated masks this small became available about 25 years ago [17].First realized by electron beam lithography they can nowadays also been inscribed using photolithography [54] as well as focused ion beam lithography [50]. Diffraction at mechanical gratingshas led to numerous experiments with atoms [17, 55], molecular dimers [8, 9] and complexhot molecules [10, 50]. Only recently we were able to demonstrate that even Nature providesnanomasks suitable for molecular diffraction in form of the silified frustule (skeleton) of algae,such as Amphipleura pellucida [56].

C. Optical gratings

The use of mechanical nanomasks with high-mass molecules is challenged by two facts: theattractive van der Waals interaction between the polarizable molecules and the dielectric ormetallic grating wall leads to a strongly velocity-dependent phase modulation of the molecu-lar matter wave [10, 57, 58]. It may even cause molecular loss for large particles that stick to thegrating walls [59]. In this case, it seems appealing to replace mechanical structures by elementsmade of light.

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1. Measurement induced absorptive gratings

Every absorptive grating may be viewed as realizing a projective measurement: the periodicarrangement of the slits in a membrane defines a transmission function for the particles to pass.If the particle does not hit the grating bars, its wave function is projected onto the comb of slitopenings.

The importance of measurement-induced gratings was described by Storey et al. [60], whosuggested that optical interactions may serve a similar purpose. For atoms, an optical maskcan be realized in the form of a standing light wave which pumps metastable noble gas atomsinto undetected states [61]. Only in the dark nodes of the optical lattice, atoms pass the gratingwithout excitation and can be detected. This idea was used in a full three-grating interferometerwith absorptive light gratings, where the atoms were optically shelved in undetected magneticstates [62]. The idea can be generalized to a much larger class of particles if we replace theresonant atom-light coupling by single-photon ionization in an ultraviolet standing light wavegrating. This proposal [31, 63] was only recently realized for the first time in our lab [64] andapplied to clusters of molecules, as discussed in Section 6.

2. Optical phase gratings

Nanomechanical and optical absorption gratings are typically accompanied by an additionalphase contribution. In mechanical structures this is caused by the van der Waals interactionbetween the molecule and the grating wall. In optical diffraction elements it arises due to theinteraction between the optical polarizability of the particle and the electric field of the lasergrating. The concept of phase gratings goes back to the idea of Kapitza and Dirac who suggestedin 1933 that electrons might be reflected at a standing light wave due to the ponderomotivepotential [65]. Diffraction of particles at light was first observed with neutral atoms [66] via thedipole interaction. The dipole force accounts for the fact that the electric laser field induces anelectric dipole moment in the particle which in turn interacts with the electric field of the laserbeam. The dipole interaction potential modulates the phase of the matter wave.

As compared to nanoparticles, atomic resonance lines are often narrow and the atom-fieldinteraction can be strongly enhanced when the detuning δ = ωL−ωA between the laser frequencyand the atomic transition frequency is small, i.e. comparable to the transition line width Γ. Theatomic response to an external light field may vary by to about five orders of magnitude acrossthe excitation spectrum with resonance widths of the order of Γ ' 10− 100 MHz. In contrast tothat, electronic transitions in complex molecules can be as broad as 1-50 THz, including a rangeof vibrational and rotational states. Even far-off resonance laser beams may still exert a force onthe particles but their magnitude is substantially smaller than in the resonant atomic case.

Pure phase gratings with little absorption were first realized with atoms [67, 68]. In complexmolecules there is often also a chance for a photon to be swallowed [69], a process not alwaysfollowed by the emission of light. In the case of the fullerenes C60 and C70, for example, theoptical excitation of a singlet state S0 → S1 is followed by a non-radiative intersystem crossingS1 → T1 with a probability greater than 99%. The triplet is supposed to live at least a dozenmicroseconds and if it decays to the ground state it does so non-radiatively [70]. Generally, inlarge molecules a great number of dissipative processes compete with electro-magnetic emission.Such internal energy conversion does not affect the coherence of the center-of-mass evolution.

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xmax d

s

x

zy

sin(θ)=λ/d

xmax ≈ sλ/d d

xmax= ht md

xmax= ht md

x

zy

Figure 2. Far-field diffraction behind a double slit leads to the sketched interference fringes. Adjacentfringe maxima are separated in momentum by multiples of the grating momentum ∆p = h/d, whichdepends on the grating period d. This alone defines the distance between the fringes after a given flighttime t. Molecules of different velocity will travel different distances, but if they are observed at the sametime, they will show the same fringe pattern.

D. Matter-wave interferometry in the time domain

Early matter-wave experiments were operated with continuous beams of electrons and neu-trons. A first time-dependent aspect of diffraction was emphasized when Moshinski [71] pro-posed the possibility of neutron diffraction at a rapidly opening shutter. This idea was realizedin 1998 [72], two years earlier in different variants with cold atoms, which could be diffractivelyreflected at time-modulated potentials of light [73, 74]. In these experiments the emphasis wasput on exploiting energy-time uncertainty in analogy to the position-momentum uncertaintyrelation in the more common diffraction in position space.

Here we focus on time-dependent (pulsed) gratings and the momentum redistribution duringthe diffraction process. In the paraxial approximation, we may restrict our considerations tothe dynamics of the one-dimensional transverse wave function, regardless of its longitudinalposition. As a function of time, the interference maxima behind a grating of period d expandlike xmax = ht/md, independently of the velocity. The latter comes into play only in stationarysetups, where the detection plane is placed at a fixed distance L = vt behind the diffractionelement. In terms of the de Broglie wavelength, the separation then reads as xmax = λdBL/d.The momentum of the particle is redistributed in units of the grating momentum pd = h/d,irrespectively of velocity or mass (see Figure 2).

Since 1991 several implementations of precision atom interferometry have followed the ex-ample of Kasevich and Chu [75] in implementing atom interferometry with pulsed Raman beamsplitters. Time-domain Talbot-Lau interferometry with laser phase gratings was later demon-strated for cold atoms [76, 77] and ultra-cold Bose Einstein condensates [78]. Most recently,we extended this concept to interferometry with clusters of molecules using pulsed ionizationgratings [64].

Pulsed optical beam splitters offer many advantages: Depending on the beam splitter mech-anism, the interference fringe shifts can be largely independent of the particle velocity. A certaindegree of selection is needed only to ensure that every detected molecule has interacted se-quentially with all three gratings. Optical gratings are widely adaptable in situ and over time.

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x1 x2 x3

z

l1l2

L L

d

Tr ansmitted Signal

Coherencepreparation Diffraction

Interferencepattern

Figure 3. Concept of Talbot-Lau interferometry: Spatially incoherent molecular beams can be broughtto interference in a three-grating setup where the first grating serves as an absorptive filter that acts asa periodic array of coherent sources. This grating can be realized by a nanomechanical mask or by astanding light wave that depletes the molecular beam via ionization or fragmentation. Diffraction at asecond grating with the same period leads to interference at the position of the third grating. The resultingmolecular density pattern has the same periodicity as the gratings; it is resolved by scanning the thirdgrating over the fringes. This setup was used in optics [79], x-ray imaging [80] and with atoms [81].Here we use it with complex molecules in three different grating configurations: with material gratings(Talbot-Lau) [49], with two material and one phase grating (KDTLI) [82], and with three pulsed ionizationgratings (OTIMA)[64].

Ionization gratings and, more generally photo-depletion gratings are rather insensitive to fre-quency fluctuations; they are rugged and indestructible. Modern lasers define gratings with aprecise periodicity and allow one to time their separation in time with nanosecond accuracy.

E. From far-field to near-field diffraction and near-field interferometry

In textbooks on classical optics the effect of diffraction is usually described in the Fraunhoferlimit of long distances behind the grating. The interference pattern can then be understood asthe Fourier transform of the aperture transmission function. In the near field [45] the theoreticaldescription involves Kirchhoff-Fresnel integrals, which account for the wave front curvature.

Far-field interference is easily understood as a wave phenomenon since the separated diffrac-tion orders constitute a clear signature of wave behavior. However, this requires a tight collima-tion which dramatically reduces the transmitted signal. In experiments on molecule diffractionless than one billionth of all emitted molecules contribute to the final interferogram [59].

Talbot-Lau (TL) near-field interferometers circumvent this problem (see Figure 3). They use atleast two, in most cases three gratings of the same grating period. The TL setting is based on theobservation by H.F. Talbot in 1836 [83] that under coherent illumination wavelength-dependentself-images will form behind a periodic structure, at multiples of the Talbot time (or Talbotlength)

TT =md2

hor LT =

mvd2

h=

d2

λdB. (1)

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This is known as the Talbot effect and was observed with collimated beams of atoms [9, 84]. Thestrong collimation requirement is alleviated in the TL setup by adding two further gratings ofequal period. The first grating acts as a mask and prepares matter-wave coherence at a distanceL, which must be at least of the order of the Talbot length. There, the second grating diffracts themolecular center-of-mass wave function. An interference pattern with the periodicity d is formedat the same distance L behind the second grating, and a third grating can be used to spatiallyprobe the molecular density pattern. The detected signal in Talbot-Lau experiments can exceedthe throughput of far-field experiments by more than a factor of ten thousand: The first gratingeffectively realizes thousands of tiny molecular beam sources, and the detection can be extendedover large areas, either by direct surface imaging or by a highly multiplexing detector, part ofwhich is the third grating in the TL-interferometer.

The TL concept has proven fruitful in optical interferometry for several decades [79]. Its gen-eralization to atom interferometry is due to John F. Clauser [81] who demonstrated the effect forpotassium atoms and also proposed to extrapolate it to ’small rocks and little viruses’ [85]. Sincethe year 2002, a series of experiments in our group at the University of Vienna has then exploredincreasingly massive and complex particles in a number of different interferometers that builton this concept [41, 86] with important adaptations. The first Talbot-Lau interferometer withthree mechanical gratings was realized in 2002 [49]. It was used for collisional [13] and thermaldecoherence [12] studies, for our first steps into biomolecule interferometry [87] and moleculemetrology [88]. The original three-grating interferometer was adapted to an interferometer withtwo mechanical gratings and one atomically clean surface detector in 2009 [42]. The strong de-phasing effects caused by van der Waals forces between the molecules and the nanogratingsmotivated us to propose [89] and realize [82] Kapitza-Dirac-Talbot-Lau interferometry (KDTLI),as described in Section 5. In order to generalize the concept of light gratings, OTIMA interfer-ometry was established as described in Section 6 [31, 63, 64].

Care has, however, to be taken: In all these cases the detection of a periodic fringe patternmight in principle also be mimicked by a moiré shadow effect [90, 91], under certain circum-stances. A detailed quantitative analysis, provided in the following section, is therefore neededto distinguish genuine quantum interference effects from classical ballistic trajectories [49, 89, 92].

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III. A UNIFIED PHASE-SPACE DESCRIPTION OF THREE-GRATING MATTER WAVE INTERFER-OMETRY

The conventional description of matter-wave interference and diffraction at fixed aperturesinvolves solving the Kirchhoff-Fresnel diffraction integral for the wave function. This is justi-fied in the presence of stationary potentials and as long as we can approximate the source asemitting pure quantum states. In most molecular quantum delocalization experiments so far,the longitudinal distance between the diffractive mask and the detection screen was also muchlarger than the size of both the aperture and the interference pattern. This permits us to changeto a time-domain treatment by eliminating the longitudinal z-coordinate. The paraxial approxi-mation allows us to describe the propagation of the transverse single-particle wave function in aco-moving frame. Diffraction at a grating structure, which varies only in the x-direction, trans-forms only the one-dimensional wave function ψ (x). In a stationary setup the propagation timet = L/vz is related to the screen distance L via the longitudinal velocity vz . One thus evaluatesthe interference pattern for fixed vz and then averages over the velocity distribution.

A. The Wigner function representation

In the following we will sketch the theory behind the Talbot-Lau interferometer (TLI) schemeusing a one-dimensional phase-space model. The latter applies to both the stationary KDTLIsetup and the pulsed OTIMA setup (as well as to the conventional TLI schemes with materialgratings not discussed here). The description is based on the Wigner function formalism [93],a most suitable and elegant representation for near-field interference phenomena. Given thedensity operator ρ that describes the reduced one-dimensional quantum state of motion in thetransverse direction, the Wigner function reads as

w (x, p) =1

2π~

∫ds eips/~〈x− s

2|ρ|x+

s

2〉. (2)

It is normalized to∫

dxdpw (x, p) = 1, a real-valued function, but not necessarily positive. Infact, its negativity implies the presence of quantum coherence. The main advantage for our pur-poses lies in its natural implementation of the quantum-classical correspondence: Not only doesit reproduce the same marginal (position and momentum) distributions as its classical coun-terpart, e.g.

∫dpw (x, p) = 〈x|ρ|x〉. The Wigner function is actually positive and identical to

the corresponding classical phase-space distribution f (x, p) for mixed states that do not exhibitquantum coherence. Moreover, w (x, p) and f (x, p) share the same time evolution along classicaltrajectories when they are subject to at most harmonic potentials [93]. Given a constant accelera-tion a, for instance, the Wigner function evolves in time by means of the shearing-displacementtransformation

wt (x, p) = w0

(x− pt

m+ a

t2

2, p−mat

), (3)

and so does ft (x, p).Putting everything together, the Wigner function is a convenient tool to assess near-field

quantum interference and to compare it with the moiré shadow effect arising from a hypotheticalclassical description. Moreover, we will see later that standard environmental decoherence andphase averaging effects are also easily incorporated into the description; they render the quantumand the classical description indistinguishable.

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B. Grating diffraction in phase space

Essentially, two ingredients are required for the coherent description of Talbot-Lau interfer-ometry in phase space: the transformation (3) corresponding to the free propagation of theparticles between the gratings and the grating transformation, which is to be discussed now.As already noted above, we can resort to an effective one-dimensional treatment of the inter-ferometer, because the relevant diffraction effects are restricted to the transverse x-coordinate ofthe grating structure. The grating transformation shall not influence notably the motion of theincident particles along the y-axis and the longitudinal z-axis. This is guaranteed in a short-timehigh-energy scattering limit, where the transverse wave function of an incident particle is subjectto a local scattering transformation of the form [94]

ψ (x) 7→ t (x)ψ (x) = |t (x)| exp [iφ (x)]ψ (x) . (4)

We denote by t (x) the complex transmission function of the d-periodic grating. Disregarding thefinite grating size, the function can be expanded into the Fourier sum t (x) =

∑n bn exp (2πinx/d).

The standard textbook example of an incident plane wave transforms into a superposition ofplane waves,

exp

(ipx

~

)7→

∞∑n=−∞

bn exp

[i

~

(p+ n

h

d

)x

], (5)

which illustrates the emergence of discrete diffraction orders shifted by multiples of the gratingmomentum h/d. The basic effect, i.e. the redistribution to different momenta, is the same bothif the grating modulates the amplitude or the phase of an incident matter wave. An absorptivemask differs from a pure phase grating merely in terms of the transmission probability, |t (x)|2 =∑

nAn exp (2πinx/d), which equals unity in the latter case.

The redistribution of momenta in steps of h/d due to diffraction also shows up explicitly inthe phase-space picture. The grating transformation (4) translates into an integral transformationof the Wigner function, w (x, p) 7→

∫dq T (x, p− q)w (x, q), with the convolution kernel [94, 95]

T (x, p) =1

2π~

∫ds eips/~t

(x− s

2

)t∗(x+

s

2

)=∑n

exp

(2πinx

d

)∑j

bjb∗j−nδ

(p− 2j − n

2

h

d

). (6)

It will be convenient to introduce the so-called Talbot coefficients at this point,

Bn (ξ) =∑j

bjb∗j−n exp [iπ (n− 2j) ξ] , (7)

and write

T (x, p) =1

2π~∑n

exp

(2πinx

d

)∫ds eips/~Bn

(sd

). (8)

14

1. Thin stationary gratings for fast particles

For stationary grating arrangements, the above local scattering transformation (4) is valid forparaxial matter-wave beams and thin gratings. To be specific, we ignore the transverse x-motionof the particles while they are passing through the grating. For the case of, say, a material gratingmask of thickness b and slit opening s = fd, and given a matter-wave beam divergence angle α,we require that b tanα/fd 1. The Fourier components of the aperture function |t (x)|2 thenread as An = Bn (0) = fsinc (πnf), with f = s/d the slit opening fraction.

In addition, the phase of the incident matter waves is modulated in the presence of an in-teraction potential V (x, z) between the particles and the grating. In the limit of high longitudi-nal velocities vz , where the kinetic energy of the particles is large compared to the interactionstrength, the modulation is approximately given by the so-called eikonal phase [94, 96],

φ (x) = − 1

~vz

∫dz V (x, z) . (9)

Note that this adds to the velocity dependence of stationary Talbot-Lau interference, where thepropagation time between the gratings is determined by L/vz . This additional dependence can,in fact, get prohibitively severe for heavy molecules in material gratings due to the strong vander Waals attraction between the particles and the grating walls. When averaged over realisticvelocity distributions this could kill the interference effect entirely [82].

The KDTLI scheme discussed here employs a standing-wave laser of wavelength λ and powerPL as a pure phase grating for polarizable nanoparticles. Given the polarizability α, the particleinteracts with the standing-wave field E ∝

√f (y, z) cos (2πx/λ) via the optical potential V =

−α |E|2 /4. This yields the phase

φ (x) = φ0 cos2

(2πx

λ

), with φ0 =

8αPLhcε0wywz

∫dzvzf (0, z) =

4√

2παPLhcε0wyvz

. (10)

We assume a Gaussian laser profile, f (y, z) = 2 exp[−2 (y/wy)

2 − 2 (z/wz)2]/πwywz , with a

waist wy that is large compared to the width of the matter-wave beam, whereas the longitudinalwaist wz shall be small (thin grating). The standing-wave grating period amounts to d = λ/2.

We find that the Fourier coefficients of the transmission function of the standing-wave phasegrating, t (x) = exp [iφ (x)], are given by Bessel functions, bn = in exp (−iφ0/2) Jn (φ0/2). Theexplicit form of the corresponding Talbot coefficients (7) follows from an addition theorem forBessel functions [97],

Bn (ξ) = Jn [ζcoh (ξ)] , with ζcoh (ξ) = φ0 sinπξ. (11)

In practice, one may need to include momentum averaging effects due to the absorption orRayleigh scattering of grating photons [92, 98]; they result in modified Talbot coefficients de-pending on the absorption and the scattering cross-section of the particles. However, these aremostly minor, negligible modifications in all the cases presented here.

2. Short ionizing grating pulses

The gratings in the OTIMA scheme are short standing-wave laser pulses, and the local scatter-ing transformation (4) holds when the particles are approximately at standstill during the pulselength τ . Given the integrated pulse energy EL =

∫τ dt PL (t), and assuming that the laser spot

15

profile is much wider than the illuminated particle ensemble [98], the eikonal phase shift can bewritten as

φ0 =4παELhcε0

f (0, 0) . (12)

The ionizing UV standing-wave pulses, however, are not just pure phase gratings. It is, in fact,required for the Talbot-Lau scheme to work that the first and the third grating are amplitude-modulating masks to establish initial coherence and to implement a position-resolving detectionof the interference pattern. Here, single-photon ionization is employed to achieve this task,as many nanoparticles, e.g. metal clusters, ionize upon the absorption of a single UV energyquantum.

We model the photon absorption as a Poisson process, where the probability of absorbing kphotons from the pulse, Pk(x) = exp [−n (x)]nk (x) /k!, is determined by

n (x) = n0 cos2

(2πx

λ

), with n0 =

4σabsELλ

hcε0f (0, 0) . (13)

The term n0 denotes the mean number of absorbed photons at the antinodes of the standingwave, with σabs the absorption cross-section of the particles. The latter is typically written asthe imaginary part of the complex dipole polarizability, χ = α + iλε0σabs/2π. For example, thecomplex polarizability of a nanosphere of radius R and dielectric permittivity ε of its materialreads as [99] χ = 4πε0R

3 (ε− 1) / (ε+ 2).Assuming that the first absorbed photon ionizes the particle, and that the ions are removed

from the ensemble, the transmission probability is given by |t (x)|2 = P0 = exp [−n (x)]. Thisleaves us with the following transmission function and its Fourier components;

t (x) = exp[(iφ0 −

n0

2

)cos2 πx

d

], bn = exp

(iφ0

2− n0

4

)In

(iφ0

2− n0

4

), (14)

with In a modified Bessel function. Once again, the Talbot coefficients (7) are obtained from anaddition theorem [98],

Bn (ξ) = e−n0/2

[ζcoh (ξ)− ζion (ξ)

ζcoh (ξ) + ζion (ξ)

]n/2Jn

[sgn ζcoh (ξ) + ζion (ξ)

√ζ2

coh (ξ)− ζ2ion (ξ)

], (15)

with ζion (ξ) = (n0/2) cosπξ and the same phase modulation term ζcoh (ξ) as in (11). The ratiobetween the amplitude and the phase modulation in the grating is conveniently described by thematerial parameter

β =n0

2φ0=

Imχ

Reχ=σabsε0λ

2πα, (16)

which does not depend on the grating intensity. The OTIMA scheme is intended for stronglyabsorbing particles with β & 1.

3. Classical pendant of the grating transformation

A fringe modulation that occurs in the particle density in the near field behind the gratingdoes not necessarily indicate quantum interference. Hypothetically speaking, shadow fringescould emerge just as well if classical particles traversed the grating on ballistic trajectories. A

16

quantitative analysis is therefore necessary to clearly distinguish true wave interference from thepredictions of a classical ballistic model.

A classical density modulation behind the grating could arise due to two effects: (i) a mod-ulation of the incident particle ensemble by absorptive grating masks blocking a fraction of thetrajectories (e.g. by the walls between adjacent slits in a material grating), and (ii) a classicallensing effect due to interaction forces exerted on the particles by the grating.

Substituting the above Wigner function (2) with the positive phase-space distribution f (x, p)of a classical particle ensemble, we can discuss the classical counterpart of the grating transfor-mation (6). The masking effect (i) is simply achieved by a multiplication of the incident particlestate with the transmission probability of the grating, f (x, p) 7→ |t (x)|2 f (x, p). For the lensingeffect (ii), on the other hand, one must in principle know the precise deflection of the trajectoriesthrough the grating. In full analogy to the above eikonal approximation (9) for the quantum case,we can approximate the particle deflection by the momentum kick q (x) = ~∂xφ (x), which repre-sents the integrated transverse force acting on the particle at position x in the grating (neglectingits motion). The momentum kick is again d-periodic and it transforms f (x, p) 7→ f (x, p− q (x)).

Putting the two effects together, the classical grating transformation can be expressed in termsof the convolution kernel

Tcl (x, p) = |t (x)|2 δ [p− ~∂xφ (x)] =1

2π~∑n

exp

(2πinx

d

)∫ds eips/~Cn

(sd

). (17)

We see that it can be brought to the same form as the quantum kernel (8), but with different,classical Talbot coefficients

Cn (ξ) =1

d

∫ d/2

−d/2dx |t (x)|2 exp

[−2πinx

d− iξd∂xφ (x)

]. (18)

It turns out that the classical terms differ from the quantum expressions (11) and (15) for thelaser gratings in the KDTLI and the OTIMA scheme by the substitution [92, 98] ζcoh (ξ) 7→ φ0πξand ζion (ξ) 7→ n0/2.

C. The Talbot self-imaging effect

With the phase-space grating transformation at hand, we can discuss the elementary near-field diffraction effect that forms the basis of the Talbot-Lau experiments studied here: the Talboteffect [79, 83]. It states that a periodic grating structure illuminated by a perfect plane waveis reconstructed in the density distribution at integer multiples of the Talbot time behind thegrating. This recurring Talbot image is the result of constructive near-field interference betweenall the outgoing wavelets diffracted at the grating.

This effect is easily assessed in phase space. Suppose that an eigenstate of zero transversemomentum, w0 (x, p) ∝ δ (p), illuminates a periodic grating t (x). The diffracted Wigner function,w1 (x, p) = T (x, p), shall then propagate freely (and without acceleration), according to Equation(3), and the matter-wave density distribution wt (x) = 〈x|ρ|x〉 =

∫dpwt (x, p) is subsequently

recorded after a time t. Using the explicit form of the grating transformation kernel (8), we find

wt (x) =

∫dp T

(x− pt

m, p

)=∑n

Bn

(nt

TT

)exp

(2πinx

d

), (19)

with TT the Talbot time as defined in (1). This density distribution describes a periodic fringe

17

time t/TT

tran

sver

se p

ositi

on x

/d

0 0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

time t/TT

0 0.5 1 1.5 2 2.5

Figure 4. Simulated density distribution as function of time t (in units of the Talbot time) and transverseposition x (in units of the grating period) in the near field behind a standing-wave phase grating (left) andbehind an ionizing laser grating (right). High density regions are dark. The gratings are illuminated bya perfectly collimated matter-wave state |p = 0〉. The phase parameter is set to φ0 = π in both cases, andβ = 1.0 is assumed for the ionizing grating. Talbot images of the grating profile occur at integer multiplesof the Talbot time.

pattern, where the time-dependent Fourier amplitudes are given by the Talbot coefficients (7) ofthe grating.

Figure 4 depicts two exemplary Talbot carpets, i.e. matter-wave density distributions behinda standing-wave phase grating, as used in the KDTLI scheme (left panel), and behind an ionizinggrating, as used in the OTIMA scheme (right panel). The Talbot effect can be observed at distincttimes behind the grating, which are given by integer multiples of the Talbot time, t = NTT . Here,the Talbot coefficients reduce to Bn (N) = (−)nN Bn (0) and the interference pattern (19) mimicsthe grating mask profile (shifted by half periods),

wNTT (x) =

∣∣∣∣t(x+Nd

2

)∣∣∣∣2 . (20)

In the case of a pure phase grating, where |t (x)|2 = 1, the Talbot images exhibit no amplitudemodulation (see left panel), while strong fringe oscillations appear in between adjacent Talbottimes.

The recurring Talbot image of the grating profile is a characteristic feature of quantum waveinterference, whereas the classical ballistic description given in sec. III B 3 leads to significantlydifferent results. If we replace the Talbot coefficients in (19) by their classical counterparts (18)for the standing-wave phase grating and for the ionizing grating, we obtain the classical Talbotcarpets depicted in fig. 5. Talbot images of the grating profile do not occur here, whereas aclassical lensing effect is clearly visible.

18

time t/TT

tran

sver

se p

ositi

on x

/d

0 0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

time t/TT

0 0.5 1 1.5 2 2.5

Figure 5. Hypothetical density distribution in the near field behind a standing-wave phase grating (left)and behind an ionizing laser grating (right) illuminated by classical ballistic particles. The same parame-ters are used as in Figure 4.

D. Talbot-Lau interference in phase space

The observation of recurring Talbot images and Talbot carpets like the ones depicted in fig. 4requires strictly coherent illumination. As in the case of far-field diffraction, matter waves mustbe collimated to less than the grating momentum h/d, which is impractical for most high-massinterference experiments. A single narrow collimation slit would simply throw away too muchof the matter-wave signal.

This problem can be circumvented by placing another grating of the same period beforethe actual interference grating1. The first grating, which must be an absorptive mask, thengenerates sufficient matter-wave coherence for the second one while transmitting a reasonablylarge fraction of the particles. In principle, the initial particle ensemble before the first gratingneed not have any transverse coherence at all; still, high-contrast interference fringes can emergeat specific distances (times) behind the second grating, as will be described in the followingbefore we move on to a realistic model of the KDTLI and the OTIMA scheme.

1. Coherent description

Let us start with a poorly collimated particle ensemble represented by an incoherent mixtureof momenta, ρ0 =

∫dpD (p) |p〉〈p|, with D (p) ≥ 0 a normalized momentum distribution. Its

characteristic width ∆p shall extend over many grating momenta, ∆p h/d, which gives acoherence length of the order of ~/∆p d and means that the state is basically unsuitable fordiffraction at a grating structure of period d. The effect of the grating is merely that of a classicaltransmission mask.

The Wigner function of the initial matter-wave state, w0 (x, p) = D (p) /∆x, is indistinguish-able from a classical particle ensemble with the same momentum distribution. For simplicity,

1 Different grating periods are also possible [94], but their ratio must be chosen carefully.

19

we neglect any fringe effects related to the finite spatial extension of the state assuming that ituniformly covers many grating periods, ∆x d. It allows us to work with strictly periodicfunctions, and the grating transformation given in Section III B maps the state into

w1 (x, p) =1

∆x

∫dq T (x, p− q)D (q) =

1

2π~∆x

∑n

e2πinx/d

∫dsBn

(sd

)D (s) eips/~. (21)

Here, the Fourier transform of the momentum distribution, D (s) =∫

dpD (p) e−ips/~, is re-stricted to arguments s ∼ ~/∆p d, which allows us to approximate the Talbot coefficientsby the Fourier components of the grating mask, Bn (s/d) ≈ Bn (0) = An. In other words, thebroad initial momentum spread is essentially unaltered by the grating and the transmitted stateis simply given by the masked Wigner function

w1 (x, p) ≈ 1

∆x

∑n

e2πinx/dAnD (p) =1

∆xD (p) |t (x)|2 . (22)

The same expression holds in the classical case, but this does not mean that no quantum coher-ence is present. In fact, the matter waves emerging from the grating slits begin to delocalize astime evolves. For the moment, let us omit acceleration in the free state evolution (3). After a timeT1, nondiagonal elements of the density matrix (i.e. spatial coherences) appear,

〈x− s

2|ρ|x+

s

2〉 =

∫dpw1

(x− pT1

m, p

)e−ips/~ =

∑n

AnD

(s+ nd

T1

TT

)e2πinx/d. (23)

This coherence function exhibits a dominant peak of magnitude |A1| at the values s = ±dT1/TT .Hence, adjacent slits of a second grating placed at T1 = TT would be coherently illuminated bythe matter waves—the basis of the Talbot-Lau effect. Clearly, the coherence is sharply limited tothose s-values where the contributions of matter waves emerging from adjacent source slits ofthe first grating add up in phase.

In a sense, each slit of the first grating acts as a single partly coherent matter-wave source for aTalbot image of the second grating [89], and it overlaps with the offset images of all other sourceslits. A visible interference pattern can be expected only at specific resonance times behindthe second grating where the offsets match the fringe oscillations. Let us work out how this isdelivered by our theoretical model.

The state (22) emerging from the first grating G1 shall evolve for the time T1 according to (3),now including the possibility for an external acceleration a 6= 0. It then transforms as (8) at asecond grating G2 of the same period, which results in the Wigner function

w2 (x, p) =

∫dq T (2) (x, p− q)w1

(x− qT1

m+aT 2

1

2, q −maT1

)=

1

2π~∆x

∑n,`

A(1)n exp

2πi

d

[(n+ `)x− naT

21

2

]

×∫

dsB(2)`

(sd

)D

(s+

nhT1

md

)exp

[i

~(p−maT1) s

]. (24)

Superscript labels (k) are used to distinguish the gratings Gk. We are looking for a fringemodulation in the probability density w3 (x) =

∫dpw3 (x, p) to find a particle at position x

20

after another time T2 behind G2,

w3 (x) =

∫dpw2

(x− pT2

m+aT 2

2

2, p−maT2

)=

1

∆x

∑n,`

A(1)n B

(2)`−n

(`T2

TT

)D

(`T2 + nT1

TTd

)

× exp

2πi

d

[`

(x− aT 2

2

2− aT1T2

)− naT

21

2

]. (25)

This is a d-periodic Fourier sum, where the sharply peaked function D restricts the Fourieramplitudes to those summation indices that fulfill |`T2 + nT1| TT . There is at most one indexn for each Fourier order ` that lies within this bound, since we require times T1,2 of the orderof the Talbot time to distinguish between quantum interference and classical shadow fringes. Apronounced fringe modulation, which is carried by the lowest harmonic orders ` = ±1,±2, . . .,appears only if the corresponding index n of the Talbot coefficients of the first grating is alsosmall.

This resonance condition selects few specific time ratios T1/T2 at which high-contrast Talbot-Lau fringes are possible; the times before and after the second grating cannot be varied inde-pendently. Consider for instance a deviation of 5% from the obvious resonance T1 = T2, sayT1/T2 = 0.95. Then the first possible Fourier contribution would come from the high-orderindex pair (`, n) = (±19,∓20), an undetectable fringe oscillation at 19 times the fundamentalfrequency!

We restrict our view here to the standard case T1 = T2 ≡ T , as used in all molecular Talbot-Lau experiments. It yields a pronounced fundamental fringe oscillation; the interference patternreduces to

w3 (x) =1

∆x

∑`

A(1)−`B

(2)2`

(`T

TT

)exp

[2πi`

d

(x− aT 2

)]. (26)

In most Talbot-Lau experiments, a movable third grating mask G3 of the same period is used incombination with a mass-spectrometric particle detection scheme in order to resolve the interfer-ence fringes. The detection signal is proportional to the fraction of particles transmitted throughG3 as a function of its lateral position xs relative to the other gratings,

S (xs) =∑`

A(1)−`A

(3)−`B

(2)2`

(`T

TT

)exp

[2πi`

d

(xs − aT 2

)]∝∫

dxw3 (x)∣∣∣t(3) (x− xs)

∣∣∣2 . (27)

The fringes are recorded by varying the lateral shift xs. Their contrast can be quantified in termsof the visibility

V =maxx S (x) −minx S (x)maxx S (x)+ minx S (x)

∈ [0, 1] , (28)

but in practice it suffices to measure the contrast in terms of the sinusoidal visibility

Vsin = 2

∣∣∣∣∣A(1)1 A

(3)1 B

(2)2 (T/TT )

A(1)0 A

(2)0 A

(3)0

∣∣∣∣∣ . (29)

21

The latter is naturally obtained as the ratio between the amplitude and the offset of a sine curvefitted to the measurement data—a more noise-robust quantity than V involving all data pointsand not only the greatest and smallest ones.

In the following, we will apply the developed model to specific Talbot-Lau schemes employ-ing optical gratings. They will differ mainly in the Talbot coefficients of the three gratings, seeSection III B. The quantum predictions are straightforwardly compared to a hypothetical classi-cal model of ballistic particles: One must merely replace the Talbot coefficients B(2)

n (ξ) of thesecond grating with their classical counterparts (18). Everything else remains the same.

We note once again that the quantum Talbot coefficients (7) are periodic in the argument ξ,whereas the classical ones in (18) are not. Specifically, we find B

(2)n (ξ) = (−)nξ A

(2)n for integer ξ,

which explains the recurring images of the second grating mask at times T = NTT .

2. The KDTLI setup

The KDTLI setup is a stationary configuration of two material grating masks G1,3 and astanding-wave phase grating G2, see fig. 9. The Talbot coefficients of the phase grating weredefined in (11), and the Fourier coefficients of the material masks are A(1,3)

n = f1,3sinc (πnf1,3),given the slit opening fractions f1,3. Due to the stationary grating arrangement, the propagationtime T = L/vz between the gratings depends on the longitudinal velocity of the particles. Theinterference signal must therefore be averaged over the velocity distribution µ (vz) of the matter-wave beam,

S (xs) = f1f3

∑`

sinc (π`f1) sinc (π`f3) exp

(2πi`xsd

)×∫

dvz µ (vz) J2`

[φ0 sin

(π`

L

LT

)]. (30)

The velocity dependence is contained in the phase modulation parameter φ0, defined in (10),and in the Talbot length LT = vzTT . External acceleration is omitted here as the gratings areoriented horizontally. In deflectometry experiments, where an electric field is applied to inducea polarizability-dependent acceleration, a velocity-dependent phase shift of the fringe patternmust be inserted again.

Note that the interference contrast vanishes at integer multiples of the Talbot length, L =NLT , when the argument of the Bessel function is zero—a consequence of the standing-wavephase grating. Maximum fringe contrast is expected to recur between two consecutive Talbotorders, depending on the strength of the phase modulation φ0. The sinusoidal visibility reads as

Vsin = 2

∣∣∣∣sinc (πf1) sinc (πf3)

∫dvz µ (vz) J2

[φ0 sin

(πL

LT

)]∣∣∣∣ . (31)

The classical shadow visibility is obtained by replacing sin (πL/LT ) with πL/LT in the argumentof the Bessel function. The expressions given here are valid for particles with vanishing absorp-tion and scattering cross-sections at the wavelength of the grating laser. The fringe visibility islower for particles that absorb or scatter a significant number of standing-wave photons.

22

3. The OTIMA setup

The pulsed OTIMA setup is a time-domain Talbot-Lau scheme, where all three gratings arerealized by ionizing standing-wave pulses, see fig. 17. Given the Talbot coefficients defined in(15), the detection signal reads as

S (xs) = exp

(−n

(1)0 + n

(2)0 + n

(3)0

2

)∑`

I`

(n

(1)0

2

)I`

(n

(3)0

2

)exp

[2πi`

d

(xs − aT 2

)]

×[ζcoh (`T/TT )− ζion (`T/TT )

ζcoh (`T/TT ) + ζion (`T/TT )

]`J2`

(√ζ2

coh

(`T

TT

)− ζ2

ion

(`T

TT

)). (32)

In the experiment, the power of each grating pulse can be varied individually, which admitsdifferent mean absorption numbers n(1,2,3)

0 , as defined in (13).In the (current) OTIMA setup all three standing-wave gratings are formed by the same mirror

surface, and so the lateral position xs of the detection grating cannot be shifted in a straightfor-ward manner. There are however means to implement and control an effective shift withoutusing an independent mirror. One practiced method is to slightly tilt the second laser beam.It leads to a negligible change in the standing-wave period that accumulates to a significantphase shift xs many wavelengths away from the mirror surface [64]. Another method wouldbe to apply an external deflection field, which yields a controllable acceleration a and gratingshift δx = aT 2. In the current experimental realization, the gratings are vertically oriented andgravitational acceleration a = g is actually present. It plays no role as long as the interfered par-ticles are small and the interference times short. Nevertheless, one could implement a significantgrating shift for heavy particles with greater Talbot times by varying the pulse delay time T .

E. The influence of environmental decoherence

The destructive influence of the environment on the Talbot-Lau interference effect is an im-portant factor to consider in the high-mass regime. Decoherence by gas collisions and by theemission and absorption of thermal radiation imposes strict vacuum and temperature condi-tions in order to observe interference fringes. These conditions are described quantitatively byincorporating the relevant decoherence processes into our phase-space model.

As far as the center-of-mass motion of particles is concerned, all relevant free-space decoher-ence effects are related to the momentum transfer in random elastic or inelastic scattering eventswith environmental degrees of freedom. Each decoherence process is characterized by an eventrate Γ (t) (which could depend on time) and by a probability distribution g (q) of momentum qtransferred in a single event. The effect on the interference pattern can be understood by simplestochastic considerations; a more formal argument is found in [95].

Suppose that a scattering event occurs at the time t ∈ [−T, T ] before or after the second gratingin a Talbot-Lau setup, and that it transfers the momentum q = (qx, qy, qz) onto the particle; onlythe x-component qx will influence the fringe pattern. The Wigner function of the particle thentransforms as w (x, p) 7→ w (x, p− qx) at the time t before (after) G2. Recalling once again thefree time evolution (3) on rectilinear trajectories in phase space, it is as if the first (third) gratingwas laterally displaced by δx = qx (T − |t|) /m. The greatest displacement occurs when theevent happens immediately at the second grating, t = 0. In Fourier space, the displacement isrepresented by phase factors exp (2πinδx/d) for each Fourier component A(1,3)

n of the first (third)grating.

23

Such displacements by scattering are now randomly distributed due to the uncontrollablenature of the interaction. We do not know the exact momentum recoil q transferred to theparticle in the scattering event, and so we must average the phase factor in Fourier space overthe distribution g (q). This results in modified Fourier amplitudes, which are all (except n = 0)reduced by a factor smaller than unity,

A(1,3)n (t) = A(1,3)

n

∫d3q g (q) exp

[2πinqxmd

(T − |t|)]

︸ ︷︷ ︸≤1

. (33)

It does not matter whether it occurs before or after the second grating in the symmetric gratingarrangement considered here.

The multiplication transformation (33) is how a single decoherence event at time t affects thefringe amplitudes in the Talbot-Lau interference signal (27). Although we cannot find out whensuch a random event happens either, we know the event rate Γ (t) and so we can model the meandecoherence effect by a simple decay process,

ddtA(1,3)n (t) = Γ (t)

∫d3q g (q) exp

[2πinqxmd

(T − |t|)]− 1

A(1,3)n (t) . (34)

The formal solution of this differential equation yields the reduction factor

Rn = exp

−∫ T

−TdtΓ (t)

[1−

∫d3q g (q) exp

(inqxd

~T − |t|TT

)]≤ 1. (35)

It represents the reduction of the nth-order Fourier amplitude of the interference fringe signal(27) by decoherence. The sinusoidal visibility (29) is reduced by R1. If there are more than oneindependent types of environmental decoherence, each contributes a factor of this form.

The most important decoherence mechanisms for hot nanoparticles are the emission of ther-mal radiation and the collision with residual gas particles. They give stringent upper boundsfor the internal particle temperature Tint and the background gas pressure pg in a Talbot-Lauexperiment [95].

As an example, let us consider the thermal radiation of a particle that is significantly hotterthan the environment, Tint Tenv. Neglecting small corrections due to the finite heat capacitanceof the particle, the spectral emission rate is given by

γemi (ω) =( ωπc

)2σabs (ω) exp

(− ~ωkBTint

), (36)

with σabs (ω) the absorption cross section for radiation of angular frequency ω. This yields thetotal emission and decoherence rate Γ =

∫∞0 dω γemi (ω), as well as the normalized and isotropic

momentum transfer distribution g (q) =(c/4π~Γq2

)γemi (cq/~).

The influence of thermal and collisional decoherence on Talbot-Lau interference is illustratedin fig. 7. The theoretical prediction fits well to the measured reduction of visibility due to anincreased background pressure or internal temperature.

F. Spontaneous collapse models

High-mass interferometry offers an ideal testbed for alternative theories on the nature ofthe quantum-classical transition. Commonly subsumed under the term macrorealism [32], they

24

suggest that standard quantum mechanics ceases to be valid and must be modified on the macro-scale in order to reconcile it with the fundamental principles of classical physics. In particular,the quantum superposition principle is to be eliminated from the macroscopic scales in order toresolve the issue of definite measurement outcomes.

The best studied of such macrorealistic hypotheses is the model of continuous spontaneouslocalization (CSL) [33, 34]. It postulates an objective modification of the Schrödinger equation ofmechanical systems by a nonlinear and stochastic term that induces a spontaneous collapse ofdelocalized matter waves above a certain mass scale. The model is characterized by essentiallytwo parameters: The atomic collapse rate λCSL, which is amplified by the total mass of a givensystem in atomic mass units, and the localization length rc down to which matter waves are col-lapsed. The latter is conventionally fixed at about rc = 100 nm, the former is currently estimatedas λCSL ∼ 10−10±2 Hz.

Fortunately, the quantitative predictions of the CSL model are easily implemented into thetheory of Talbot-Lau interferometry. In fact, the observable consequences of the CSL modificationin center-of-mass experiments with nanoparticles smaller than rc mimic those of a ficticiousdecoherence process, as discussed in Section III E. The decoherence rate parameter is simplysubstituted by Γ = (m/1 u)2 λCSL, and the momentum transfer distribution g (q) by a symmetricGaussian distribution of standard deviation σ = ~/

√2rc [100]. The sinusoidal Talbot-Lau fringe

contrast (29) would be reduced by the factor [31]

Vsin 7→ Vsin exp

−2( m

1 u

)2λCSLT

[1−√πrcTTdT

erf(

dT

2rcTT

)](37)

due to CSL. We notice a quadratic mass dependence in the exponential decay rate. Moreover,the grating separation time T , which must be of the order of the Talbot time TT , must also beincreased in proportion to the particle mass. All this allows us to test the CSL predictions byobserving high-contrast interference with heavy particles, placing an upper bound on the CSLrate parameter λCSL with each successful experiment.

25

IV. TALBOT-LAU INTERFEROMETRY

Interferometers of the Talbot-Lau geometry have various advantages over simple far-fielddiffraction schemes with a single grating: First, they are more compact due to the less stringentcollimation requirements. In practice, the TLI design is about an order of magnitude shorterthan its far-field counterpart for the same de Broglie wavelength. This entails that the requiredcoherence time can be shorter by the same factor and it renders near-field experiments lesssensitive to external perturbations. Alternatively, a near-field scheme can operate with largergratings at equal machine length, which makes it less sensitive to dispersive van der Waals inthe grating slits.

Second, since the first grating in TL interferometry comprises thousands of parallel coher-ence preparation slits, it increases the signal throughput by four to five orders of magnitude incomparison to a far-field setup with a single pair of collimation slits. However, the alignmentprocedure of three-grating interferometers are clearly more demanding. The interference fringevisibility strongly depends on a precise alignment of the gratings with respect to each other andwith respect to external force fields. Given the large grating period and the even larger supportstructure, alignment of these grating could be conveniently done by comparing laser diffractionimages.

Molecular Talbot-Lau interferometry was first demonstrated with thermal beams of C60 andC70 [49], and soon extended to the biodyes tetraphenylporphyrin (TPP) and to the larger fluo-rofullerenes (C60F48) [87]. The setup is sketched in fig. 6. Three gold gratings with a period of990 nm, an open fraction of f = 0.48 and a thickness of t = 500 nm were originally manufac-tured by Heidenhain in Traunreut/Germany for use on the X-ray satellite AXAF/Chandra. Eachof them was photolithographically written into a gold membrane that spanned a free circle of16 mm diameter. Each membrane stretched across a thin steel ring which was then magneticallyattached to the grating mount to avoid mechanical stress. They were positioned in a mutualdistance first of 23 cm [49] and in later experiments at 40 cm for explorations of longer coherencetimes and higher interference orders [101].

Fullerenes were efficiently detected using delayed thermal ionization after interaction with afocused strong laser beam (here 10-30 Watt of 514 nm or 532 nm light), whereas other molecules,which do not photoionize, were detected by means of electron impact ionization and quadrupolemass spectrometry in the detector stage. Since the velocity distribution of thermal sources iswell described by a Maxwell-Boltzmann distribution, in some cases also with a small velocityoffset [46], the molecular coherence length was improved by distinguishing the longitudinalvelocities in the beam according to different flight times, which correspond to different free-fall distances in the gravitational field. This allowed us to establish a selectivity of ∆v/v '15− 20% (FWHM).

A. Protection from collisional and thermal decoherence

All initial experiments were targeted at obtaining the best possible quantum coherence andthe highest fringe visibility. In the given setup, the width of the slit openings and the finite ve-locity spread limited the interference contrast to about 30%, both in experiment and in theory. Avariety of different effects may reduce the visibility even further: Some causes may appear ’triv-ial’, such as vibrational disturbances of the entire interferometer or the individual gratings [102],but they often decide success or failure in practice. Vibrations on the hull of the vacuum chamberas tiny as 15 nm were observed to translate into sizeable perturbations inside the interferometer,at specific vibrational frequencies.

26

fullerenesource

lateral limiter

1. grating

2. grating

ionisationlaser

detector

3. grating

gold gratingsperiod 1 mμ

L1

L2

height limiter

shift [µm]

0

32

1Imin

Imax

Figure 6. Setup of the first Talbot-Lau interferometer for molecules: a thermal source of fullerenes wascollimated to 2 × 0.2 mrad2 to limit the alignment requirements and to allow a gravitational velocityselection by three vertical height delimiters (oven nozzle, slit, detecting laser beam waist). The moleculeswere sent through a sequence of three microstructured gold gratings (d = 990 nm). One expects and findsa sinusoidal interferogram. The fringe visibility is then readily defined by the modulation amplitude to thesignal offset. A quantitative comparison of the observed fringe visibility with the predicted dependenceon the de Broglie wavelength then allows to confirm the true quantum nature of this molecular densitypattern. This requires also to take into account that the attractive van der Waals interaction between thepolarizable electron cloud of a fullerene molecule and the gold grating bar modulates the matter-wavephase and reduces the effective slit opening. [49].

On the more foundational side, we may also ask how the delocalized particles may dis-seminate which-way information into the environment. This happens through the momentumexchange when the particles either collide with rest gas molecules in the high-vacuum chamberor when they emit thermal photons, as discussed in Section III E. Decoherence theory accountsfor these effects and helps in understanding the transition from quantum to classical behaviour.

The quantum decoherence experiments in Vienna were focused on two questions: First, whichpressure can still be tolerated in the interferometer chamber before quantum coherence is washedout? Second, what molecular temperature is still allowed before the thermal emission of radi-ation becomes predominant and a severe obstacle to the observation of molecular quantuminterference? Exemplary results are plotted in fig. 7.

From the exponential decrease of the fringe visibility with increasing rest-gas pressure (seefig. 7a) one may deduce the cross section for collisional decoherence assuming that none ofthese collisions is sufficiently violent to remove the molecules from the detected particle beam.We found that rest gas pressures of 10−6 mbar are clearly too high for observing interference oflarge molecules on the time scale of milliseconds. One can show that a technologically achievablepressure of 10−10 mbar would suffice to interfere particles beyond 106 u in the future [31].

The dependence of quantum coherence on the internal temperature was studied by exposingthe fullerenes to intense laser light. Every single absorbed photon contributed to the internalheating of the molecule. Although this process may lead to molecular ionization–a fact thatwas used in the first C60 diffraction experiments–the emission of thermal radiation is the faster

27

0 2 4 6 8 10

1540 2580 2880 2930 2940

0 4 8 12 160,0

0,2

0,4

0,6

0,8

1,0

Mean microcanonical Temperature (K)

b)a)

Heating Laser Power (W)

N

orm

alis

ed V

isib

ility

Pressure (10 -7 mbar)

Figure 7. Experimental decoherence in Talbot-Lau interferometry: a) Interference contrast of fullerenesas a function of the residual gas pressure. The exponential decrease is in agreement with the theoreticalpredictions based on the expected collisional cross sections. The small-angle van der Waals cross section,which is relevant for gas collisions, may exceed the geometrical particle cross section by two orders ofmagnitude [13]. b) Interference fringe visibility as a function of the internal molecular temperature: Eachfullerene molecule has a high number of internal degrees of freedom and acts as its own internal heatbath. Upon emission of a visible or near-infrared photon which-path information becomes available andrandom momentum kicks lead to a reduction of the interference contrast [12].

cooling process at high temperatures. If a visible or near-infrared photon is emitted by theinterfering molecule in free flight between the gratings, its spatial coherence should be reducedby the degree of which-path information disseminated into the environment.

Again, extrapolation to complex particles shows that high-mass matter-wave interferenceshould still be possible, provided we can cool the particles to the temperature of liquid nitrogenor ideally even liquid helium, where all electronic and vibrational degrees of freedom are frozeninto their ground states. The remaining rotational excitations will not harm the experiments,since the associated decay rates are extremely small and since the wavelength of a rotationalphoton is far too long to provide any information about the molecular position.

B. Quantum-assisted deflectometry

The idea of quantum-assisted deflectometry builds on established classical beam methodswhich were successfully applied to characterize beams of clusters [103–106] and molecules [107,108]: A homogeneous force field Fx = αstat (E · ∇)Ex deflects a particle of static molecularpolarizability αstat by the amount

∆x = αstat(E · ∇)Ex

mv2z

s(s

2+ l). (38)

Here E is the applied electric field, m is the molecular mass, vz the longitudinal velocity, sdesignates the length of the deflection electrode, and l the distance of the electrode from G3.

This deflection can be measured with nanoscale precision in the Talbot-Lau near-field schemesince molecular interferogram serves as a tiny ruler. Lateral shifts as small as 10 nm can still be

28

Figure 8. Quantum assisted deflectometry for polarizability measurements in Talbot-Lau interferometry:A pair of electrodes is designed to create a particularly homogeneous electric force field on polarizablemolecules. In devices with a gravitational velocity selection scheme one usually operates with a rectangu-lar molecular beam that is broader than high. This requires a suitable electrode design which ensures therequired homogeneity of the deflection force field down to the level of 1% [109]. The voltage dependentdeflection of the molecular interferogram then allows one to extract the polarizability. This idea has beenused to measure the polarizability of the fullerenes C60 and C70 [88].

resolved, while most of the classical machines would have a ten thousand times lower resolutionwhen they operate with a collimated beam. The interferometric deflectometer was first demon-strated to retrieve the static polarizabilities of C60 and C70. In fig. 8 the experimental setup isshown as well as a sketch of the fringe shift in the external field.

29

Source G1Chopper

λ/2 platePBS532 nmP<18 W

Laser

G2 G3

Source chamber Interferometer chamber

--

-

----

Detection chamber

EIionisation

QMSLN2 BaffleD1 D2 D3

x

zy

Figure 9. Sketch of the KDTL interferometer. The molecules are evaporated in an oven, velocity selectionis performed by three height delimiters D1, D2, and D3. The molecules traverse tree gratings - G1, G2,and G3 - all having an equal periodicity of 266 nm. G1 and G3 are SiN gratings acting as absorptivemasks for coherence preparation and detection, respectively. G2 is a standing light wave produced byretro-reflection of a 532 nm laser at a plane mirror acting as the diffraction grating. G3 is moved laterallyacross the molecular density pattern in order to scan the interference fringes. The molecules transmittedthrough G3 are ionized by electron ionization and detected in a quadrupole mass spectrometer.

V. KAPITZA-DIRAC-TALBOT-LAU (KDTL) INTERFEROMETRY

In Kapitza-Dirac-Talbot-Lau (KDTL) interferometry the overall concept, beam sources, inter-ference scanning and detection are done in close analogy to the TL-interferometry describedabove [49]. A key difference between the two devices is that the mechanical diffraction gratingG2 is substituted by a standing light wave (λL = 532 nm, cw) which acts predominantly as aphase mask. This eliminates the strongly position- and velocity-dependent phase modulationthat we observed in TL-interferometry because of van der Waals forces in this central element.In addition, in our implementation [82] the mechanical masks in G1 and G3 are substantiallythinner (t = 160 nm instead of 500 nm) and finer (d = 266 nm) than in the previous TL interfer-ometer. They are written into silicon nitride which is substantially stiffer than gold and also lesspolarizable. In all KDTLI experiments performed so far the molecules were again evaporated ina stable thermal source. A sketch and photograph of the interferometer are shown in fig. 9 andfig. 10.

A. Experimental results: high-mass quantum interference

The KDTL interferometer has served us in many studies with about a dozen of different or-ganic molecules. Initially demonstrated for the fullerene C70 and a perfluoroalkyl-functionalizedazobenzene [82], it has proven to be well suited for quantum experiments with substantiallylarger compounds, up to perfluoroalkyl-functionalized nanospheres [111] and porphyrin deriva-tives [22].

A major contribution to the success of all these experiments was the ability of modern chem-istry to provide tailored functional components. Already the earlier experiments with fluoro-

30

Source Chamber Interferometer Chamber Detection Chamber

Effusive Source

Nitrogen Baffle

Differential Pumping

Chopper

Valve

Interferometer

DeflectionElectrodes

QMS

Figure 10. Experimental setup of the KDTL interferometer. The vacuum chamber is mounted on anoptical table for vibration isolation. It is divided into three segments for the source, the interferometer,and the detection chamber. Collisional decoherence is avoided by pumping the entire system to about10−8 mbar.

fullerenes had indicated that a high fluorine content is beneficial for the volatilization of massivecompounds. It generates stable intramolecular bonds while lowering the overall molecular polar-izability. This reduces the intermolecular attraction and thus the temperature required to achievea sizeable vapor pressure. This makes it possible that even molecules in the mass range of manythousand atomic mass units can still form intense and stable molecular beams [110, 113].

A most recent addition to the set of functionalized molecules is C284H190F320N4S12. It is builtaround a tetraphenylporphyrin core to which a number of perfluoroalkyl side chains were at-tached. It is generally demanding to synthesize particles with a well-defined number of sidechains so that a ’library’ of molecules with well-defined mass differences emerges quite natu-rally in the synthesis of these complex compounds. The fact that it contains different numbersof side chains can actually be useful when substituting quadrupole mass spectrometry by time-of-flight detection, as used in OTIMA interferometry, where the comparison of different masspeaks is done in parallel. We have studied quantum interference specifically for the twelve-chain compound which was post-selected from the molecular beam by electron impact ioniza-tion quadrupole mass spectrometry after interference. Each particle has a molecular weight of10 123 u and consists of 810 atoms. An illustration of one of its possible 3D structures is shownin fig.11m).

Figure 12 shows the best interference pattern obtained for this molecule. In order to dis-tinguish the experimental molecular density pattern from a possible classical shadow pattern(a phase-grating variant of a Moiré pattern [90]) we plot also the observed fringe visibility asa function of the diffracting grating laser power which imprints a phase onto the matter wavewhen it passes the anti-nodes of the standing light wave. The observed fringe visibility is in

31

10 Å

a b c d e

f

g

h i

j k

m

l

Figure 11. Selection of molecules, which showed high-contrast quantum interference in our near-fieldmatter wave interferometers: a) and b) The fullerenes C60 and C70 were used in far-field diffraction [10],TL interferometry [49] and KDTL interferometry [82]; c) & d) the fluorofullerene C60F48 showed interfer-ence in a TL setup [87] but its polar variant C60F36 only revealed good quantum interference in KDTLIseveral years later [92]; e) the perfluoroalkylated nanosphere C60[C12F25]8 has a high vapor pressureand therefore guided the way to high-mass interference with thermal molecular beams [110, 111]; f) ex-periments with perfluoroalkyl-functionalized azobenzenes C30H12F30N2O4 demonstrated that quantuminterference persists even under conformational state changes in free flight and interference-enhancedstudies could quantitatively assess the conformational dynamics [40]; g) the tailor-made molecule C25H20

and its larger derivative h) C49H16F52 served to reveal the emergence of vibrationally induced dipolemoments in floppy molecules [112]; KDTL interference was capable of distinguishing the structural iso-mers h) & i) in quantum-interference enhanced deflectometry; j) Tetraphenylporphyrin C44H30N4 whichis a common dye molecule [87] showed quantum coherence and in comparison with its derivatives k)C44H28ClFeN4 and l)C44H28FeN4 it allowed us to study the influence of a permanent dipole moment onmatter-wave coherence [112]; m) the functionalized porphyrin C284H190F320N4S12 is currently the mostmassive molecule to show high-contrast quantum interference [22].

32

0 100 200 300 400 500 600 700 800 900 1000 11000

400

800

1200

1600

2000

G3 position (nm)

sign

al (3

s)-1

(a)

(b)

0 0.5 1 1.5 20

10

20

30

40

50

P ower (W )

Visi

bilit

y (%

)

Figure 12. Quantum interference of the functionalized porphyrin C284H190F320N4S12 ([22] shown in figure11 m)),currently the most massive and complex particle to show quantum interference. It consists of810 atoms per particle and has a mass of 10 123 u. (a) Matter-wave interference pattern recorded at a laserpower of P ∼= 1 W. The circles show the experimental signal as a function of the position of G3 with theshaded area corresponding to the detection background signal. The blue solid line represents a sinusoidalfit to the data resulting in a fringe visibility of 33 %. A classical picture predicts a visibility of only 8%for the same experimental parameters. (b) Plot of the measured interference visibility as a function ofthe laser power in the standing light wave. The theoretically expected contrast according to the modelin [92] is plotted as the blue (quantum) and orange (classical) lines respectively. The dashed blue linescorrespond to the expected quantum contrast for an increased (reduced) mean velocity of 5 m/s.

good agreement with a full quantum model, as derived in Section 3 [92], and markedly differentfrom the classical expectations [112].

Mass and atom number of our custom-made particles are already comparable to thoseof small proteins such as insulin (5 700 u) or cytochrome C (12 000 u). We have opted forperfluoroalkyl-functionalized synthetic structures in all of our earlier experiments and stillin many ongoing studies, since fluorination enables the volatilization of massive particles underconditions where proteins or DNA would naturally denature or fragment.

33

0 2 4 6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Laser Power (W)

Visi

bilit

y

Figure 13. from [92]: Measured power dependence of the quantum interference visibility for C60 (bluedots) and C70 (orange dots). The solid lines represent the quantum expectation for fitted α532nm andσ532nm. The dashed lines represent the corresponding classical predictions.

B. Experimental results: quantum-interference assisted metrology

Quantum interferometry with large molecules is highly sensitive to external forces, since thede Broglie wavelength is very small. Due to the small grating periodicity in our the KDTLinterferometer, fringe shifts smaller than 10 nm can still be resolved. This makes quantum-assisted metrology a powerful tool for measuring inertial forces and even more so for revealinginformation on internal molecular properties, including electric, magnetic or optical propertiesas well as internal structure and dynamics.

1. Optical polarizability

The fringe visibility depends on various parameters (Eq. (10)): the molecular optical polar-izability α532nm determines the strength of the phase modulation φ0 in the standing light waveand hence the interference visibility (see Equation (31)). In fig.13 the power dependence of theinterference of C60 and C70 is displayed. When all other experimental parameters are known,in particular the velocity distribution, the laser beam waist, the grating properties and the in-terferometer geometry, the value of α532nm can be determined from the power dependence ofthe fringe visibility. For a pure phase grating, where photo absorption in G2 can be neglected,a single-parameter fit suffices to extract this value with an accuracy of a few percent. In thepresence of finite absorption, the associated cross-section σ532nm can be fitted additionally as asecond free parameter [92]. The strong influence of the particle polarizability on the fringe con-trast can be used to complement mass spectrometry [39]. In a first demonstration experimentwe asked for instance whether molecular fragmentation of a complex chemical rather occurs inthe thermal source under the influence of heat or in the mass spectrometer as a result of electron

34

0 2 4 6 8 10 12 14

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Laser Power (W)

Visi

bilit

y

Figure 14. KDTL interference with C96H48Cl2F102P2Pd (m=3379 amu) allows one for instance to decidewhether molecular fragmentation occurs in the source (thermally) or in the detector (by electron impact).The interaction with the diffracting standing light wave depends crucially on the optical polarizabilitywhich is different for the parent molecule and its two equal fragments, although their polarizability-to-mass ratio is the same [39].

impact ionization. While standard mass spectrometry would only see the final fragment, theorigin of the fragmentation process determines whether the relevant de Broglie wavelength λdB

and particle polarizability α532 in the interferometer are associated with the fragment (in our ex-ample: C48H24F51P, m = 1601 u) or the parent molecule (here: C96H48Cl2F102P2Pd, m = 3379 u).A measurement of the quantum visibility as a function of the laser power allows us to clearlydecide whether the observed interference pattern is due to the parent molecule or one of itsfragments (see fig.14). This is difficult to achieve in standard mass spectrometry. Even dedicatedexperiments in physical chemistry and classical beam deflectometry would only be sensitive toαstat/m which is the same for the two compounds [114].

2. Static polarizability

Similar to Talbot-Lau interferometry before (Section 4), the KDTL apparatus is also equippedwith a high voltage electrode for quantitative matter-wave deflectometry and it allows us toaddress a much wider class of molecules. The absence of the van der Waals interaction in G2 isof particular importance for experiments with polar molecules.

3. Vibration induced electric dipole moments

We have already seen that a molecular interference pattern may shift in response to an ex-ternal electric field and that for non-polar rigid molecules, such as the fullerenes, this responseis well described by their static polarizability αstat. In addition to that, floppy molecules may

35

Figure 15. Snapshots of a molecular dynamics simulation for the functionalized azobenzene displayed inFigure 11 f). Conformation changes on the nanosecond time scale cause the electric dipole moment to varyby 300%, even though the average dipole moment of the ground state configuration vanishes. Changes inthe nuclear positions leave the molecular polarizability largely invariant since the overall electronic bondstructure is also largely unaffected [40].

undergo thermally-induced conformational changes which in general also entail the develop-ment of electric dipole moments which may fluctuate on the nanosecond time scale. The in-teraction with the external field is then appropriately described by the electric susceptibilityχ = αstat +

⟨d2x

⟩/kBT [115] which also includes the thermal average of all squared dipole com-

ponents along the field axis⟨d2x

⟩. The susceptibility replaces αstat in Equation (38).

This phenomenon was observed and verified with perfluoroalkyl-functionalized azobenzenes.Molecular dynamics simulations by N. Doltsinis in Münster showed that these molecules un-dergo many conformational changes and develop non-zero electric dipole moments when theyare heated to a temperature of 400 K. Some snapshots of exemplary configurations are shown infig.15. The resulting fringe shift in an external field agreed well with the expectations from themolecular dynamics simulation [40].

For molecules g) and h) in fig. 11 we also compared the relative rigidity of substructuresinside the delocalized molecules. The molecules share an identical core: TetraphenylmethaneC25H20 (shown as compound g) in dig. 11) and the enlarged derivative C49H16F52 (molecule h)in fig. 11) equipped with four floppy perfluoroalkyl chains. In consistence with our earlier resultswe found that for the small tetraphenylmethane molecule the electric susceptibility is dominatedby the static polarizability. For the larger, floppy derivative the effective susceptibility is not

36

0 2 4 6 8 10 12 140

5

10

15

20

25

30

35

Laser Power (W )

Visi

bilit

y (%

)

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

35

Voltage (kV)

Vis

ibili

ty (%

)

a b

Figure 16. From [112]: a) Interference contrast as a function of the laser power in the standing lightwave for FeTPP (orange squares, molecule k) in fig. 11) and FeTPPCl (blue dots, molecule l) in fig. 11).Due to their comparable mass, optical polarizability, mean velocity, and velocity spread they behave verysimilarly in the interferometer when no deflection voltage is applied. b) Quantum contrast as a function ofelectric deflection voltage. The (nearly) non-polar FeTPP (orange squares) is only slightly phase-averagedbecause of the finite velocity spread in the molecular beam. The polar FeTPPCl (blue dots) is highlysensitive to the external field as differently oriented molecules will also be deflected in different directionsby different amounts.

sufficiently approximated by the static polarizability–vibrationally induced dipole moments playa significant role.

The experiments on floppy molecules show clearly that violent internal dynamics is fullycompatible with high contrast de Broglie coherence as long as it does not provide any meansto retrieve which-path information. In this case the center-of-mass motion is clearly decoupledfrom the internal state. At the same time, the internal properties lead to a measurable shift of thecenter-of-mass wave function. This is actually the principle behind quantum-assisted metrologyin our case.

The important factor for the preservation of coherence is the absence of correlation betweenthe internal and the external motion. Even in the presence of the external field there is no way toassign a specific path to the molecule. The internal state is important but it remains separable.It contributes only to the interaction potential that determines the overall molecular evolution.

4. Permanent electric dipole moments

In contrast to non-polar molecules which maintain their quantum contrast even in the pres-ence of comparably strong electric fields, we observe fringe averaging for a rigid, polar com-pound already at moderate fields (see fig.16). This can be demonstrated using two molecularvariants which are very similar in mass and polarizability but different in their static dipole mo-ments. The polar FeTPPCl differs from the almost non-polar FeTPP – see fig. 11 k) and l) – by asingle chlorine atom which causes an electric dipole moment of approximately 2.7 Debye [116].Since both molecular species are very similar in mass, beam velocity and optical polarizability,their interferograms look very similar (fig. 16 a) and they shift by the same amount in the sameelectric field. We observe, however, rapid fringe averaging already at moderate field values inthe interference of the polar compound (fig. 16 b). This is consistent with the view that a thermalbeam source delivers molecules in random orientations and with random directions for theirrotation axes. In the presence of the outer field the interferograms will be shifted depending onthis orientation and the total contrast averages out.

37

The experiments mentioned here give only a snapshot of all what has been done and whatcan still be done in the future. KDTL interferometry has proven to be applicable to a wide classof atoms, organic and inorganic molecules and nanoparticles, imposing only a few constraints:The particle’s optical polarizability at 532 nm needs to be sufficiently large for the laser in G2

to imprint a phase modulation on the transiting de Broglie wave of the order of φ = π. Thisis for example achieved for a molecular velocity in the range of 200 m/s, an optical polariz-ability of α532 nm = 4πε0 × 100Å3 as well as a laser power of 10 W focused in a beam waist of20µm×1000µm. Also, the absorption cross section at 532 nm needs to be sufficiently small thatthe phase grating character dominates the diffraction process. For the geometry of our experi-ment this is warranted for an absorption cross section in the range of σ532 nm ' 10−17 cm2 andthe concept works generally as long as α532 nm/σ532 nm ≥ ε0 × 10−6 m.

Since the KDTLI is an interferometer in real space rather than in the time-domain, copingwith velocity dispersive phase shifts is a general challenge. With progress in molecular beamsource, cooling and detection techniques this challenge can, however, be largely alleviated. Theconcept has been predicted to be operational for particles in the mass range of millions of atomicmass units. Even the fact that G1 and G3 are material masks does not prevent the concept fromworking in principle. As long as the particles are still transmitted, KDTLI is a viable universalinterferometer concept also for high masses.

38

VI. OTIMA INTERFEROMETRY

Talbot-Lau interferometers have proven to be suitable devices for quantum experiments withcomplex molecules, but it remains an outstanding goal to investigate the validity of quantumphysics for truly macroscopic particles. Future matter-wave interferometers should thereforefulfill certain criteria: they should be applicable to different types of particles. They should beable to cope with dispersive and therefore contrast-diminishing interactions between the particlesand their environment. They should be robust against vibrations and ideally be fitted with adetection scheme that is sensitive on the few-particle level. Our optical time domain matterwave interferometer (OTIMA) incorporates these ideas and has demonstrated its functionalityin first proof-of-principle interference experiments with clusters of organic molecules [64]. Asketch of the OTIMA setup is shown in fig. 17 and a photo in fig. 18. It is divided into threeparts: a molecular beam source, the interferometer area and a detector.

The central element of the OTIMA setup is a flat CaF2 mirror. It retroreflects three VUVlasers pulses (λ =157 nm, E < 5 mJ, τ = 7 ns) to form three standing waves with a mutualpulse delay T which should be of the order of the Talbot time. A burst of neutral particles issent in close proximity across the mirror surface while being subjected to this pulse series. Thesmall grating period of d = λ/2 = 78.5 nm results in a Talbot time of 15 ns/u, and ensures thateven particles of m = 106 u only require an unperturbed coherence time of 30 ms. The lasergratings interact with the matter waves in two different ways: On the one hand they have aphase component, as already discussed in the previous section. On the other hand they actas absorptive spatial filters for neutral particles, provided that single-photon ionization, or anyother ’depletion’ mechanism, is sufficiently probable in their anti-nodes. A photon energy of7.9 eV suffices to ionize a large class of nanoparticles, such as metal and semiconductor clusters,certain molecular clusters and some biomolecules. While G2 could also be a pure phase grating,G1 and G3 need to be absorptive masks (as in TLI and KDTLI) to imprint the required transversecoherence onto the cluster beam and to reveal the emerging interference pattern.

Interferometry in the time domain eliminates a large range of dispersive phase shifts andone may conceive arrangements where a particle cloud even expands isotropically in all direc-tions [76], with velocities of opposite sign. This is possible since grating diffraction imparts afixed transverse momentum onto the molecule. In practice, a certain velocity selection is oftenneeded to ensure that each particle sees every grating pulse. For this reason the OTIMA inter-ferometer is paired with a pulsed cluster source as well as a pulsed detector. In our currentrealization with anthracene molecules, adiabatic expansion from an Even-Lavie valve [117] isused to produce a supersonic beam of cold neutral molecular clusters. The molecules are heatedclose to their melting point inside a noble gas environment (typically argon or neon) at high-pressure. When the valve opens, it releases a burst of molecules which cool in the presenceof the co-expanding seed gas to clusters. The short (less than 30µs) dense cluster pulse entersthe interferometer region via a skimmer as well as a horizontal and a vertical collimation slit.Behind the interferometer, the remaining neutral particles are post-ionized by another VUV laserbeam and they are detected in a time-of-flight mass spectrometer (TOF-MS). This mass-resolveddetection scheme allows us to compare the interferograms of all cluster masses simultaneously.

39

Grating laser 3157nm

H

L

MCP

Grating laser 2157nm

p2

W

P

MirrorV

Grating laser 1157nm

p3

t1 t2 t3

TOF-MS

p1

Figure 17. Sketch of the OTIMA setup. A pulsed source (V) emits clouds of cold neutral particles. Thisparticle beam is delimited in height (H) and width (W) by two adjustable slits. Three nanosecond laserpulses, which are separated in space, are back-reflected by a single ultra flat mirror and form standinglight waves that ionize particles in the anti-nodes. They act on the particles with a well-defined timesequence (at t1 = 0, t2 = T and t3 = 2T ). The remaining neutral particles are ionized by a pulsed laser(L) and they are detected in a time-of-flight mass-spectrometer (TOF-MS).

A. Experimental design and conditions

1. Requirements on mirror flatness and spectral purity of the grating lasers

The dielectric surface of the interferometer mirror provides the boundary condition for allthree standing light waves and thus determines their relative phases. As long as the mirrorsurface is smooth and flat any given separation of the laser pulses will result in a well-definedand predictable phase between the position of the molecular density pattern and G3. Localcorrugations on the Angstrom level and an overall flatness of 5 nm across a mirror surface of50 mm would be ideally required.

A global mirror curvature caused by gravitational or thermal sag complicates the interferenceexperiments as the local grating shift is then correlated with the particle cloud position below

40

Figure 18. Photo of the OTIMA setup: High vacuum is a prerequisite for all matter-wave interferometers.It is established by a set of turbomolecular pumps with vibration isolation bellows. The Even-Lavie [117]cluster source emits a thermal cloud of anthracene molecules, which cluster in the adiabatic co-expansionwith an intense jet of noble gas. Top: the time-of-flight mass spectrometer is a central element, allowingto record interference with clusters of many masses simultaneously.

the mirror. Even greater constraints are imposed by mirror corrugations that exceed about 5 nmsurface modulation over the width of a single laser beam (10 mm). Since particles at differentpositions are then subjected to different grating phases the interference contrast will be dimin-ished. This also sets stringent requirements on the thermal stability of the mirror. CaF2 exhibitsminimal absorption at 157 nm and heating by the laser pulses can be neglected. On the otherhand, it is possible to illuminate the mirror locally with an intense laser at a different wavelengthto expand the material and to thermally scan the mirror phase.

At VUV wavelengths even the best available mirrors are limited to a reflectivity of R ' 96%.This entails that the transmitted fraction of the laser light (4%) cannot contribute to the formationof the standing wave. The clusters are thus illuminated by a running wave background whichreduces the useful signal as it ionizes all clusters with a small but mass dependent probability,independent of their position in the grating. This requires particular attention in the numericalsimulations of the experiments.

As in TL and KDTL interferometry it is also important in OTIMA to match the grating periodsof all three gratings over the entire width of the molecular beam. We use three F2 excimerlasers that emit predominantly on two lines, one at 157.63 nm and one at 157.52 nm, wherethe latter is about ten times weaker [118]. Both lines are about 1 pm wide and the resultinglongitudinal coherence length of about 1 cm requires the particles to pass in close proximity(∼ 1 mm) to the mirror surface. While the wavelength of many other excimer lasers, such asArF or XeCl laser, can be varied externally, F2-lasers exploit a molecular transition which varieswith pressure and temperature by less than the emission line width [118]. This is the reason whythree independent gas lasers are capable of generating three equally periodic light structures and

41

thus work together to form a high-contrast matter-wave interferometer. As in the considerationof single laser coherence, the grating nodes and antinodes of two subsequent laser beams onlyrun out of phase when we compare more than 100 000 grating periods, i.e. on the scale of severalmillimeters.

The transverse coherence of fluorine lasers is typically limited to 40 − 80µm. While at firstglance this may seem to compromise the quality of a standing light wave across its 10 mm width,it is the smoothness of the mirror surface which defines the wave fronts. Transverse coherenceis only required to ensure that a standing wave can actually be formed. The formation of astanding light wave in a millimeter distance to the mirror surface is then still guaranteed as longas the divergence or the angle of incidence is limited to below 5 mrad.

2. Vacuum requirements

Since collisions and interactions between interfering particles and background gas cause de-coherence, the interferometer area of the OTIMA is surrounded by high vacuum of p2 < 10−8

mbar. At this pressure even long ranging interactions between particles with a permanent orinduced dipole moment are minimized to a level at which coherence times of up to 30 ms withmasses beyond 106 u are reasonable [31]. In order to establish the high vacuum, the sourcechamber with a working pressure of p1 < 10−5mbar is separated from the interferometer bytwo differential pumping stages. All chambers are pumped by turbomolecular pumps that areseparated from the chamber by dedicated vibration isolation bellows.

3. Vibrational isolation

In most interferometers vibrations are an important source of dephasing [102]. They result inuncontrolled and time dependent grating shifts which eliminate the interference contrast whenthey approach half the grating period d/2. In our OTIMA setup the interferometric stability isgreatly enhanced by the fact that all three grating laser pulses are reflected by the same mirror.The total interference fringe shift in a three-grating interferometer is determined by the positionshift of each individual grating. This is true for all our Talbot interferometers as well as for manyof the atom interferometers described in this book:

∆φ = (2π/d)(∆x1 − 2∆x2 + ∆x3). (39)

Common mode perturbations, such as linear (∆x1 = ∆x2 = ∆x3) or torsional pendulum oscilla-tions (e.g. ∆x2 = 0,∆x3 = −∆x1) cancel out if they occur on a time scale longer than the transittime through the interferometer. Hence, low frequency mirror tilts and shifts can be neglected.In OTIMA interferometry, the time scale is set by the Talbot time TT and thus by the particle’smassm. Mirror displacements larger than 10 nm at a frequency greater than 2π/TT , will how-ever cause vibrational dephasing and loss of interference contrast. In our current experiment, themirror vibrations were measured to be smaller than ∆xi ≤ 5 nm within about 25µs, i.e. on thetime scale of the Talbot time for anthracene clusters. This can be measured in situ with an opticalMichelson interferometer. For experiments with higher mass particles, this optical readout cancross-correlated with the interference signal to correct for vibrations [119].

42

4. Alignment to gravity and the rotation of the Earth

The reflective surface of the OTIMA interferometer mirror faces downwards and the gratingaxes are parallel to Earth’s gravity. The vertical acceleration of the clusters does not lead toany dispersive phase shift since all particles stay in the interferometer for the same amount oftime and fall by the same distance. This alleviates the alignment requirements with respect togravity and will be important in tests of the equivalence principle with masses ranging fromsingle atoms to clusters of 106 u, in the future [120, 121]. The gravitational deflection can evenserve another purpose: it yields an effective fringe shift ∆x = gT 2, which should the same forall particles if the equivalence principle is valid. It can therefore be used to implement a spatialscan of the interference pattern by varying the pulse delay T . A limit is, however, reached whenthe particles are deflected to beyond the longitudinal coherence length of the grating lasers. Thislimits our present interferometer to m < 106 u, since at 2TT = 30 ms all particles fall by 4 mm.

In addition to that, the rotation of the Earth will induce a Coriolis phase shift ∆φ = 4πs ·(v ×Ω)T 2/d which cannot be eliminated even in a pulsed setup. The normal vector to the mirrorsurface s can also be aligned parallel to the axis of Earth’s rotation to eliminate the Coriolis shiftto first order. Higher order effects, caused by the beam splitting process and gravitational freefall remain as minor contributions [122–124].

5. Beam divergence

In the ideal case of perfect timing and negligible grating pulse duration, the molecular beamdivergence has no influence on the interference contrast. In reality, however, the pulse lengthof the grating lasers is limited to about 8 ns and the resolution of our timing jitter monitor(δt = 1 ns) sets an upper bound to the timing precision of the gratings. This is relevant be-cause an asymmetry in the pulse delays causes a timing mismatch between the formation andthe probing of the interference pattern. Although each particle interferes only with itself, allparticles must constructively contribute to the same interference pattern. For molecules arrivingat G3 under different angles this only works if the timing jitter of G3 and the angular spreadof the particle trajectories are small. With increasing divergence α the averaging over relativelyshifted interference patterns leads to a reduction of the overall interference contrast. The beamdivergence therefore imposes an upper bound on ∆T . The visibility vanishes completely if thefringe pattern is averaged over half the grating period d,

T

L⊥<

∆T

d/2→ ∆Tmax =

d

2v tanα↔ ∆T

T<

1

2N, (40)

where L⊥ = Nd is the distance that the most divergent particle with longitudinal velocity vflies in the direction of the grating during time T (see fig.19). For a particle with a longitudinalvelocity of 1000 m/s and a divergence angle of 1 mrad, the interference pattern will vanish oncethe two pulse delays T3 − T2 and T2 − T1 differ by more than 40 ns. This can be seen in fig. 19where the interference contrast is plotted as a function of the asymmetry in the pulse delays, asdescribed in Sec. VI B 2.

B. Experimental results

All experimental data shown in this section have been recorded with clusters of anthracenemolecules. Anthracene (C14H10) is an aromatic hydrocarbon with a mass of 178 u and its struc-

43

T

L =

T v

tan

α

= N

d

α

∆T

d/2

t

x

d

a

T1 T2 T3

0 40-40 80-80

0

0.2

0.4

0.6

0.8

∆S

of A

c7

Detuning ∆T [ns]

Tim

e se

quen

ceN

48 ns

resonant∆T = 0

o - resonant∆T ≠ 0t1 = 0 t2 = T t3 = 2T + ∆T

t1 = 0 t2 = T t3 = 2T

b

Figure 19. Influence of the beam divergence on the required timing precision. a) Particles that travel underdifferent angles form the same interference patter - but only if both pulse separation times are equal. If thethird grating pulse arrives too early or late, the particles interfering along the upper or lower trajectorieswill contribute to interference patterns with a relative shift. b) In the anthracene cluster experiment thebeam divergence sets an upper limit of 40 ns to the imbalance of pulse separation times. Since the laserjitter is fixed in the experiments but the Talbot-time grows with mass, the relative timing requirementsbecome less stringent for large particles.

ture is shown as an inset in fig. 20. It has a high vapour pressure and therefore easily formsclusters in a supersonic expansion beam, in our case produced by the Even-Lavie valve. As formany clusters, the single-molecule ionization energy of 7.4 eV decreases with the cluster size.In general, only little is known about optical properties of materials at 157 nm. This applies, inparticular to organic substances and to clusters of anthracene. The data therefore needs to beextracted from our own experiments.

1. Quantum interference seen as a mass-dependent modulation of cluster transmission

For sources that emit a broad mass distribution of clusters, an interference pattern can be ob-tained by choosing a fixed grating pulse delay and recording the interferometer transmission asa function of mass. Two different pulse timing conditions are compared to get an unambiguoussignature of quantum interference: In the resonant mode, the delays between two subsequentlaser pulses are equal to better than 1 ns , ∆T1−∆T2 ≡ (t2−t1)−(t3−t2) < 1 ns. For masses withT ' n · TT (with n ∈ N) constructive and destructive quantum interference should modulate thetransmitted cluster signal SR, depending on the grating phases. If the resonance condition is notmet, ∆T1 6= ∆T2, we do not expect any phase-dependent enhancement or reduction of the massspectrum. We therefore compare the resonant signal to the off-resonant mode where the latterone is realized by shifting the delay of the third laser so that the relative grating delays differ by200 ns. Even such a small imbalance in the pulse separations suffices to wash out the interferencepattern, when the cluster beam divergence exceeds 0.2 mrad. The ’off-resonant’ mass spectrumis thus a suitable reference SO. In comparison to SR it has the same overall transmission but itlacks the interferometric modulation of the mass spectrum.

In fig. 20 we plot the two signals as a function of mass and see a clear difference. To quantifythe interference contrast, the normalized signal difference ∆SN = (SR − SO)/SO is introduced

44

0

0.4

0.8

∆S

Ac5

Ac8

ExperimentQuantum predictionClassical prediction

Sign

al [a

.u.]

aN

20

40

60

0x7

Mass [amu]

Ac8

x10

Mass [amu]

0

0.4

0.8

50

100

0

b

535 714 892 1070 1248 1426 1605 1784535 714 892 1070 1248 1426 1605 1784 1962 2141 2319

Figure 20. OTIMA interference depends on the cluster mass and the Talbot time. The observed modula-tion on the mass spectrum peaks at the cluster mass of the anthracene decamer for v = 690 m/s while itshifts to the heptamer for v = 925 m/s. The resonances in the mass spectrum agree with the predictionsof quantum mechanics. Also the observed transmission amplitudes agree with our expectations underreasonable assumptions about the cluster polarizability and absorption cross section.

and compared to the quantum theoretical predictions. Figure 20 shows two measurements whichdemonstrate in particular the role of the pulse separation time T . We can increase the mostprobable cluster velocity by changing the co-expanding seed gas from argon to the lighter noblegas neon, while keeping the source temperature constant. This allows us to shift the interferencemaximum from the 10-fold anthracene cluster Ac10 for T = 25.2µs at a velocity of v ' 690 m/sto the heptamer Ac7 for T = 18.9µs and v ' 925 m/s. Theory and experiment are in goodagreement. For details on the theoretical model we refer the reader to Section 3 as well as to theliterature [64, 98].

2. Interference resonance in the time domain

We can visualize the matter-wave resonance also in the time domain by systematically scan-ning the asymmetry ∆T = ∆T1−∆T2 in the off-resonant mode. Figure 19 shows the interferencecontrast of the anthracene heptamer Ac7 as a function of ∆T for a fixed pulse separation time ofT = 18.9µs. For a cluster velocity of v = 960 m/s, we can extract the beam divergence accord-ing to equation (40). We find a divergence angle of α ' 0.9 mrad which agrees well with ourexpectations for the given geometry of the experiment.

3. Interference pattern in position-space

In OTIMA interferometry, a simple scan of the third grating, as done in TLI and KDTLI,is impeded by the fact that all three gratings are reflected off the same rigid mirror. We mayhowever scan the interference pattern by changing the periodicity of one of the gratings and

45

0

0.2

∆SN

0

0.2

-0.2

0.4

-0.1

-0.1

0

0.2

∆SN

∆SN

0 1000 2000 3000Distance [μm] Distance [μm]

0 1000 2000 3000

Ac4 Ac5

Ac6 Ac7

Ac8 Ac9

≙39.25nm

Figure 21. Quantum interference in position space for the Anthracene tetramer (Ac4) to the nonamer(Ac9). When the laser beam in G2 is tilted by 5 mrad, a change of periodicity in the standing light wave byseveral picometers leads to an accumulated shift of the total grating phase which varies with the distancebetween the cluster beam and the mirror surface. Retracting the mirror surface by one millimeter thereforeallows us to effectively sample the molecular density pattern in real space with a precision and resolutionof several nanometers.

by retracting the mirror vertically to move the cluster position in the light field. In spite ofthe fact that the laser frequency is fixed, its standing wave period can be altered by tilting thegrating laser beam by an angle, here specifically G2 by θ = 5.1 mrad. While the grating vectorremains defined by the orientation of the mirror surface, increasing the tilt angle changes thenormal wave vector kp = k cos θ. Many periods away from the mirror surface, in a distance ofroughly 1.5 mm, even a difference in the grating periods as small as a few picometers sums upto a relevant phase difference of the second grating with respect to the other two. Since theposition of G2 influences the total fringe shift twice as strong as the values in G1 and G3, evena phase shift of π/2, corresponding to an effective shift of only 20 nm is sufficient to switch thesignal from constructive to destructive interference. By continuously increasing the separationof the mirror and the cluster beam we can observe the expected sinusoidal transmission curve,as shown in fig. 21 for different cluster masses. The overall damping of the curves is consistentwith the finite coherence length of the VUV lasers and the vertical extension of the cluster beam.

The OTIMA interferometer is consistent with our theoretical expectations in all three aspects:We see a modulation of the transmitted mass distribution, which agrees also with the expec-tations of de Broglie waves assigned to each cluster mass at the known velocity. We observea narrow resonance in the time domain, which illustrates the potentially high sensitivity of a

46

pulsed interferometer scheme. And finally we see that also time domain interferometry creates amolecular density pattern in real space, which is as tiny as 80 nm in our current implementation.Exploiting higher-order Talbot-fringes, it is however also conceivable to create structures with aperiodicity of 40 nm or smaller, with feature sizes of less than 20 nm wide.

47

VII. PERSPECTIVES FOR QUANTUM DELOCALIZATION EXPERIMENTS AT HIGH MASSES

Even though quantum mechanics has proven to be perfectly valid in all experimental tests sofar, the number of quantum superposition experiments with increasingly macroscopic mechan-ical systems grows rapidly, opening a wide field of research at the interface between quantumand classical physics. Whether quantum theory must be modified in the macroworld is the es-sential question in this field [32, 34], with possible links to gravity theory [35–37]. Each newquantum experiment may then serve as another step closer to the answer by confirming the va-lidity of quantum mechanics in a given setting of unprecedented macroscopicity. This requires,however, a unified quantitative notion of macroscopicity that allows one to compare differentquantum experiments by the degree to which they pin down that answer and probe the validityof quantum mechanics.

Judging from the variety of quantum experiments with mechanical systems, different criteriacould be invoked to characterize macroscopic quantum effects: Nanomechanical oscillators intheir vibrational ground state count among the most massive objects ever brought to the quan-tum regime [125, 126]. On the other hand, if quantum superposition states were to be observedwith such systems, their coherent delocalization would be limited to rather microscopic sizes: thewidth of motional ground-state wave packet is rarely larger than several dozen femtometers.

In contrast to this, neutron interferometers hold the current record with regard to the areaspanned by the interference paths of a single quantum particle. This can be as large as 100 cm2,i.e. in principle wide enough to let a single neutron be delocalized around the thickness ofan arm [127]. Nowadays, atom interferometers reach a high metrological sensitivity and anenclosed interferometer area in the range of square centimeters, too [123, 124]. Modern atominterferometers employing optical beam splitters achieve superposition states with a momentumseparation of up to 102~k [128]. State of the art macromolecule interferometers, as presentedhere, operate with smaller interferometer areas but at substantially higher mass.

Another macroscopic quantum phenomenon occurs in superconducting ring experiments,where superposition states of counterrotating persistent currents involving billions of electronscan be observed [129, 130]. These superpositions exhibit a truly macroscopic difference in themagnetic moment associated to the superimposed loop currents. At the same time however, amany-body analysis shows that the two branches of the superposition differ by only a smallnumber of electrons in the end [131].

In order to combine the various aspects of macroscopicity, and in order to compare the var-ious experiments with mechanical systems, we have proposed a unified method to measuremacroscopicity [132]. It quantifies the range of macrorealistic modifications of quantum theorythat is ruled out by a given experiment. Such modifications induce classicality by adding a termto the Schrödinger equation, which collapses delocalized wave functions at a rate that ampli-fies with the system size. We have specified the mathematical form of a broad generic class ofsuch modifications and defined a logarithmic quantity µ for the macroscopicity of mechanicalquantum experiments. The microscopic reference value µ = 0 would be achieved by keeping anelectron in a superposition state for one second. Figure 22 shows the results for a representativeselection of matter-wave experiments with neutrons, atoms, SQUIDs, and molecules. The highestmacroscopicity to date, as reached in the KDTLI experiment with perfluoroalkyl-functionalizedmolecules [22, 111], is the equivalent of a single electron kept in a superposition state for morethan 1012 seconds.

Future experiments with nanoclusters in the OTIMA scheme may yield much higher macro-scopicity values of the order of 20, similar to what would be reached in other ambitious pro-posals to observe the quantum behaviour of nano- and micrometer-sized objects [133, 134]. Inthis regime, the most prominent example of a collapse modification, the model of continuous

48

neutronsatomsmoleculesSQUIDs

year of publication

mac

rosc

opic

ity µ

1960 1970 1980 1990 2000 20104

6

8

10

12

14

Figure 22. Macroscopicities for a selection of neutron, atom, molecule and SQUID superposition ex-periments ordered according to their publication date. See [132] and references therein for details onhow the data points were obtained. The highest achieved macroscopicity values are shared by molecularTalbot-Lau interferometers and high-precision atom interferometers. SQUID experiments rank low dueto the small mass of the superimposed electrons and due to the short coherence times achieved in theexperiments.

spontaneous localization (CSL) [33, 34], should become experimentally accessible. Specifically,the CSL model predicts a mass-dependent reduction of Talbot-Lau interference contrast, as dis-cussed in sec. III F. We have shown that OTIMA interference with gold clusters of about a millionatomic mass units would test the CSL model in its current formulation [31]. That is to say, theobservation of high interference visibility would place an upper bound for the rate at which theCSL effect occurs, according to eq. (37), which is comparable to the current estimate of the CSLrate parameter λCSL ∼ 10−10±2 Hz.

All this confirms that de Broglie interferometry with molecules and nanoparticles remains apromising route towards testing quantum mechanics at the borderline to the classical macroworld.The quest for higher masses has led us away from elementary double-slit textbook interferome-ters to conceptually more advanced near-field methods, which are well understood theoreticallyand which facilitated the current mass and macroscopicity records in interferometry.

On the experimental side, an important key to all future experiments remains the develop-ment of suitable sources of very massive, very slow, cold and controlled nanoparticles. Firstsuccess in feedback laser cooling [51] and cavity cooling [52, 53] of particles between 70 nm and1000 nm, i.e. with masses between 108 u and 1012 u raise hope that high-mass quantum inter-ference experiments will become available in a mass range where fascinating new physics lurkaround the corner. Well-adapted interferometer technologies are now already available.

49

ACKNOWLEDGMENTS

We thank our collaboration partners for making nanoparticle matter-wave optics an excitingadventure. We thank the Austrian Science Fund, FWF for financial support in the projects Z149-N16 (Wittgenstein) and DK CoQuS W1210-2 as well as the Ministry of science for support in theBM:WF project IS725001. We acknowledge funding by the European commission in the projectEU NANOQUESTFIT(304886), the ERC Advanced Grant PROBIOTIQUS(320694) and the ViennaZIT communication project (957475).

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